## Mc Graw Hill Key To Algebra Book 4 Polynomials 1st Edition Chapter 6 Factoring Out A Common Factor

**Page 11 Exercise 1 Answer**

Given expression is 6a−30

We find factors of the given polynomial

We take out common from the given expression 3 goes into both 6 and 30, so we factor out 3

6a − 30 = 3(2a−10)

The factorization of the given polynomial is 3(2a−10)

**Page 11 Exercise 2 Answer**

Given expression is 6a−30

We find factors of a given polynomial

We take out common from the given expression.

6 goes into both 6,30, we take that out as common 6a − 30 = 6(a−5)

The factorization of the given polynomial is 6(a−5)

**Page 11 Exercise 3 Answer**

Given expression is 6a−30

We find factors of a given polynomial

We take out common from the given expression.

2 goes into both 6,30, we take that out as common 6a − 30 = 2(3a−15)

The factorization of the given polynomial is 2(3a−15)

**Page 11 Exercise 5 Answer**

Given expression is 18x−27

We find factors of a given polynomial

We take out common from the given expression.

9 goes into both 18,27, we take that out as common 18x − 27 = 9(2x−3)

The factorization of the given polynomial is 9(2x−3)

**Page 11 Exercise 6 Answer**

Given expression is 4x−32

We find factors of a given polynomial

We take out common from the given expression.

4 goes into both 4,32, we take that out as common 4x − 32 = 4(x−8)

The factorization of the given polynomial is 4(x−8)

**Page 11 Exercise 7 Answer**

Given expression is 8y−10

We find factors of a given polynomial

We take out common from the given expression.

2 goes into both 8,10, we take that out as common 8y − 10 = 2(4y−5)

The factorization of the given polynomial is 2(4y−5)

**Page 11 Exercise 9 Answer**

Given expression is 5x − 5y + 10z

We find factors of a given polynomial

We take out common from the given expression.

5 goes into both 5,10, we take that out as common 5x − 5y + 10z = 5(x−y+2z)

The factorization of the given polynomial is 5(x−y+2z)

**Page 11 Exercise 10 Answer**

Given polynomial is 7x − 7y − 7z

We find factors of a given polynomial

We factorize the given expression.

HCF of 7x,7y,7z is 7,we take that out as common 7x − 7y − 7z = 7(x−y−z)

Factorization of the given polynomial is 7(x−y−z)

**Page 11 Exercise 12 Answer**

Given polynomial is 2x^{2 }+ 18x + 14

We find factors of a given polynomial

We factorize given polynomial

HCF of 2x^{2},18x,14is 2,we take that out as common 2x^{2} + 18x + 14 = 2(x^{2}+9x+7)

Factorization of the given polynomial is 2(x^{2}+9x+7)

**Page 11 Exercise 14 Answer**

Given polynomial is 6a − 18b + 12c

We find factors of the given polynomial

We factorize the given expression.

HCF of 6a,18b,12cis 6, we take that out as common.

6a − 18b + 12c = 6(a−3b+2c)

The factorization of given polynomial is 6(a−3b+2c)

**Page 11 Exercise 15 Answer**

Given polynomial is 9x + 12y + 15

We find the factors of a given polynomial

We factorize the given expression.

HCF of 9x,12y, and 15 is 3, we take that out as common.

9x + 12y + 15 = 3(3x+4y+5)

The factorization of the given polynomial is 3(3x+4y+5)

**Page 11 Exercise 16 Answer**

Given polynomial is 8a + 12b + 4

We find the factors of a given polynomial

We factorize the given expression.

HCF of 8a,12b, and 4 is 4, we take that out as common.

8a + 12b + 4 = 4(2a+3b+1)

The factorization of the given polynomial is 4(2a+3b+1)

**Page 12 Exercise 1 Answer**

Given the polynomial to factorize: 15y + 5

Now:

We have to factorize this polynomial by factoring out the biggest terms out of the polynomial as per Sandy’s way.

Given polynomial: 15y+5=5(3y+1)

Now we see that:

15y+5=5(3y+1)

⇒ \(\frac{15 y}{5}=3 y\)

\(\frac{5}{5}=1\)Here 5 is the biggest single term that can be taken out as a factor as it is the greatest term that is a factor of 15y,5.

The polynomial after factoring out the biggest term is as follows: 15y + 5 = 5(3y+1)

**Page 12 Exercise 2 Answer**

Given expression: 7a + 7

Now:

We have to factorize this polynomial by factoring out the biggest terms out of the polynomial as per Sandy’s way.

Given expression: 7a+7

Now we see that:

7a+7=7(a+1)

⇒ \(\begin{aligned}

& \frac{7 a}{7}=a \\

& \frac{7}{7}=1

\end{aligned}\)

Here 7 is the biggest single term that can be taken out as a factor as it is the greatest term that is a factor of 7a,7

The polynomial after factoring out the biggest term is as follows: 7a + 7 = 7(a+1)

**Page 12 Exercise 4 Answer**

Given expression: 5 + 20x

Now:

We have to factorize this polynomial by factoring out the biggest terms out of the polynomial as per Sandy’s way.

Given expression: 5+20x

factorizing the expression:

5+20x=5(1+4x)

Now we see that:

⇒ \(\begin{aligned}

& \frac{20 x}{5}=4 x \\

& \frac{5}{5}=1

\end{aligned}\)

Here 5 is the biggest single term that can be taken out as a factor as it is the greatest term that is a factor of 5,20x

The polynomial after factoring out the biggest term is as follows: 5 + 20x = 5(1+4x)

**Page 12 Exercise 5 Answer**

Given expression: 6x − 2

Now:

Given expression: 6x-2

Factorizing the expression

6x-2=2(3x-1)

Now we see that:

\(\begin{aligned}& \frac{6 x}{2}=3 x \\

& \frac{2}{2}=1

\end{aligned}\)

Here 2 is the biggest single term that can be taken out as a factor as it is the greatest term that is a factor of 6x,2

The polynomial after factoring out the biggest term is as follows: 6x − 2 = 2(3x−1)

**Page 12 Exercise 6 Answer**

Given expression: 14n + 21p + 7

Now:

Given expression: 14n+21p+7

Factorizing the expression:

14n+21p+7=7(2n+3p+1)

Now we see that:

⇒ \(\begin{aligned}

\frac{14 n}{7} & =2 n \\

\frac{21 p}{7} & =3 p \\

\frac{7}{7} & =1

\end{aligned}\)

Here 7 is the biggest single term that can be taken out as a factor as it is the greatest term that is a factor of 14n,21p,7

The polynomial after factoring out the biggest term is as follows: 14n + 21p + 7 = 7(2n+3p+1)

**Page 12 Exercise 7 Answer**

Given expression: 9x − 18y + 3

Factorizing the expression

9x-18y+3=3(3x-6y+1)

**Now we see that: **

⇒ \(\begin{aligned}

& \frac{9 x}{3}=3 x \\

& \frac{18 y}{3}=6 y \\

& \frac{3}{3}=1

\end{aligned}\)

Given expression

Here 3 is the biggest single term that can be taken out as a factor as it is the greatest term that is a factor of 9x,18y,3

The polynomial after factoring out the biggest term is as follows: 9x − 18y + 3 = 3(3x−6y+1)

**Page 12 Exercise 8 Answer**

Given expression: 11 − 22a + 44b

Now:

Given expression: 11-22a+44b

Factorizing the expression:

11-22a+44b=11(1-2a+4b)

Now we see that:

& \frac{11}{11}=1 \\

& \frac{22 a}{11}=2 a

\end{aligned}\)

⇒ \(\frac{44 b}{11}=4 b\)

Here 11 is the biggest single term that can be taken out as a factor as it is the greatest term that is a factor of 11,22a,44b

The polynomial after factoring out the biggest term is as follows: 11 − 22a + 44b = 11(1−2a+4b)

**Page 12 Exercise 9 Answer**

Given expression: 5x^{2} + 10x + 5

Now:

Given expression: 5×2+10x+5

Factorizing the expression:

⇒ \(\begin{aligned}

5 x^2+10 x+5 & =5\left(x^2+2 x+1\right) \\

& =5\left(x^2+x+x+1\right) \\

& =5(x(x+1)+1(x+1)) \\

& =5(x+1)^2

\end{aligned}\)

we see that:

⇒ \(\begin{aligned}

\frac{5 x^2}{5} & =x^2 \\

\frac{10 x}{5} & =2 x \\

\frac{5}{5} & =1

\end{aligned}\)

Here 5 is the biggest single term that can be taken out as a factor as it is the greatest term that is a factor of 5x^{2},10x,5

The polynomial after factoring out the biggest term is as follows: 5x^{2} + 10x + 5 = 5(x^{2}+2x+1)

**Page 12 Exercise 10 Answer**

Given expression to factorize: 28x* ^{2}* + 28x + 7

Now:

Given expression: 28×2+28x+7

Factorizing the expression:

28×2+28x+7=7(4×2+4x+1)

=7(4x²+2x+2x+1)

=7(2x(2x+1)+1(2x+1)

=7(2x+1)²

We can see that:

⇒ \(\begin{aligned}

& \frac{28 x^2}{7}=4 x^2 \\

& \frac{28 x}{7}=4 x \\

& \frac{7}{7}=1

\end{aligned}\)

Here 7 is the biggest single term that can be taken out as a factor as it is the greatest term that is a factor of 28x^{2},28x,7

The polynomial after factoring out the biggest term is as follows: 28x^{2} + 28x + 7 = 7(4x^{2}+4x+1)

**Page 12 Exercise 1 Answer**

Given expression: 24x + 28

We need to factor out the biggest term from this expression.

Factoring out the biggest term from the expression:

Given expression: 24x+28

factorizing the expression:

24x+28=4(6x+7)

now we see that:

⇒ \(\begin{aligned}

& \frac{24 x}{4}=6 x \\

& \frac{28}{4}=7

\end{aligned}\)

Here 4 is the biggest single term that can be taken out as a factor as it is the greatest term that is a factor of 24x,28

The polynomial after factoring out the biggest term is as follows: 24x + 28 = 4(6x+7)

**Page 12 Exercise 3 Answer**

Given expression: 40n − 24

We need to factor out the biggest term from this expression.

Factoring out the biggest term from the expression:

Given expression: 4 on -24

Factorizing the expression:

4 on-24= 8(5n-3)

now we see that:

⇒ \(\begin{aligned}

& \frac{40 n}{8}=5 n \\

& \frac{24}{8}=3

\end{aligned}\)

Here 8 is the biggest single term that can be taken out as a factor as it is the greatest term that is a factor of 40n,24

The polynomial after factoring out the biggest term is as follows: 40n − 24 = 8(5n−3)

**Page 12 Exercise 5 Answer**

Given:

The polynomial is 18x − 30y.

To find:

The objective is to find the greatest factor among the given polynomials.

Consider the given polynomial,

18x − 30y

= 6(3x−5y) Taking GCF

Therefore, the required result is 18x − 30y = 6(3x−5y).

**Page 12 Exercise 6 Answer**

Given:

The polynomial is 100 + 40z.

To find:

The objective is to factor the greatest number among the given polynomials.

Consider the given polynomial,

100 + 40z

= 20(5+2z) Taking GCF

Therefore, the required result is 100 + 40z = 20(5+2z).

**Page 12 Exercise 7 Answer**

Given:

The polynomial is 56x^{2} + 42.

To find:

The objective is to factor the greatest number among the given polynomials.

Consider the given polynomial,

56x^{2} + 42

= 14(4x^{2}+3) Taking GCF

Therefore, the required result is 56x^{2} + 42 = 14(4x^{2}+3).

**Page 12 Exercise 8 Answer**

Given:

The polynomial is 20x + 60y − 100z.

To find:

The objective is to factor the greatest number among the given polynomials.

Consider the given polynomial,

20x + 60y − 100z

= 20(x+3y−5z) Taking GCF

Therefore, the required result is 20x + 60y − 100z = 20(x+3y−5z).

**Page 12 Exercise 9 Answer**

Given:

The polynomial is 14a − 12b + 6c.

To find:

The objective is to factor the greatest number among the given polynomials.

Consider the given polynomial,

14a − 12b + 6c

= 2(7a−6b+3c) Taking GCF

Therefore, the required result is 14a − 12b + 6c = 2(7a−6b+3c).

**Page 12 Exercise 10 Answer**

Given:

The polynomial is 24 + 48x + 42x^{2}.

To find:

The objective is to factor the greatest number among the given polynomials.

Consider the given polynomial,

24 + 48x + 42x^{2}

= 6(4+8x+7x^{2}) Taking GCF

Therefore, the required result is 24 + 48x + 42x^{2} = 6(4+8x+7x^{2}).

**Page 12 Exercise 11 Answer**

Given:

The polynomial is 50a − 20b + 30c.

To find:

The objective is to factor the greatest number among the given polynomials.

Consider the given polynomial,

50a − 20b + 30c

= 10(5a+2b+3c) Taking GCF

Therefore, the required result is 50a − 20b + 30c = 10(5a+2b+3c).

**Page 12 Exercise 12 Answer**

Given:

The polynomial is 6c^{2} + 27c − 15.

To find:

The objective is to factor the greatest number among the given polynomials.

Consider the given polynomial,

6c^{2} + 27c − 15

= 3(2c^{2}+9c−5) Taking GCF

Therefore, the required result is 6c^{2} + 27c − 15 = 3(2c^{2}+9c−5).

**Page 12 Exercise 13 Answer**

Given the polynomial

12r + 36s − 60t

Here it is asked to factor out the biggest number.

We know that the factors of 12 are

12 = 1,2,3,4,6,12

and we know that the factors of 36 are

36 = 1,2,3,4,6,9,12,18,36

and we know that the factors of 60 are

60 = 1,2,3,4,5,6,10,12,15,20,30,60

So here there are 6

common factors for 12,36,60.

That is

1,2,3,4,6,12

Hence here the greatest common factor is 12

So we can write that

12r + 36s − 60t = 12(r+3s−5t)

Hence the biggest number that can be factored out is 12.

Therefore, the biggest number that can be factored out is 12.

**Page 13 Exercise 1 Answer**

Given the polynomial

16x − 48

Here it is asked to find the greatest common factor.

We know that the factors of 16 are

16 = 1,2,4,8,16

The factors of 48 are

48 = 1,2,3,4,6,8,12,16,24,48

So here there are 5 common factors for 16 and 48.

That is

1,2,4,8,16

Hence the greatest common factor of 16 and 48 is 16.

That is

16x − 48 = 16(x−3)

Therefore, the greatest common factor of 16x − 48 is 16.

**Page 13 Exercise 2 Answer**

Given the polynomial

30x + 45

Here it is asked to find the greatest common factor.

We know that the factors of 30 are

30 = 1,2,3,5,6,10,15,30

And we know that the factors of 45 are

45 = 1,3,5,9,15,45

So here there are 4 common factors for 30 and 45.

That is

1,3,5,15

Hence the greatest common factor of 30 and 45 is 15.

That is

30x + 45 = 15(2x+3)

Therefore, the greatest common factor of 30x + 45 is 15.

**Page 13 Exercise 4 Answer**

Given the polynomial

200x + 80y − 120z

Here it is asked to find the greatest common factor.

We know that the factors of 200 are

200 = 1,2,4,5,8,10,20,25,40,50,100,200

The factors of 80 are

80 = 1,2,4,5,8,10,16,20,40,80

The factors of 120 are

120 = 1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120

So here there are 8 common factors for 200,80,120.

That is

1,2,4,5,8,10,20,40

Hence the greatest common factor of 200,80 and 120 is 40.

That is

200x + 80y − 120z = 40(5x+2y−3z)

Therefore, the greatest common factor of 200x + 80y − 120z is 40.

**Page 13 Exercise 5 Answer**

Given the polynomial

72a − 96b − 48c

Here it is asked to find the greatest common factor.

We know that the factors of 72 are

72 = 1,2,3,4,6,8,9,12,18,24,36,72

The factors of 96 are

96 = 1,2,3,4,6,8,12,16,24,32,48,96

The factors of 48 are

48 = 1,2,3,4,6,8,12,16,24,48

So here there are 8 common factors for 72,96,48 are 1,2,3,4,6,8,12,24

Hence the greatest common factor of 72,96,48 is 24.

That is

72a − 96b − 48c = 24(3a−4b−2c)

Therefore, the greatest common factor of 72a − 96b − 48c is 24.

**Page 13 Exercise 6 Answer**

Given the polynomial

32x^{2} + 40x + 160

Here it is asked to find the greatest common factor.

We know that the factors of 32 are

32 = 1,2,4,8,16,32

The factors of 40 are

40 = 1,2,4,5,8,10,20,40

The factors of 160 are

160 = 1,2,4,5,8,10,16,20,32,40,80,160

So here there are 4 common factors for 32,40,160.

That is

1,2,4,8

Hence the greatest common factor of 32,40,160 is 8.

That is

32x^{2} + 40x + 160 = 8(4x^{2}+5x+20)

Therefore, the greatest common factor of 32x^{2} + 40x + 160 is 8.

**Page 13 Exercise 1 Answer**

Given the polynomial

x^{2} + 3x

Here it is asked to find the common factor.

Here we can see that x is common to both terms.

That is

x^{2} + 3x = x(x+3)

Hence the common factor is x.

Therefore, the common factor is x.

**Page 13 Exercise 2 Answer**

Given the polynomial

5a^{2} + 2a

Here it is asked to find the common factor.

Here we can see that a is common to both terms.

That is

5a^{2} + 2a = a(5a+2)

Hence the common factor is a.

Therefore, the common factor is a.

**Page 13 Exercise 3 Answer**

Given the polynomial

y^{2} − 7y

Here it is asked to find the common factor.

Here we can see that y is common to both terms.

That is

y^{2} − 7y = y(y−7)

Hence the common factor is y.

Therefore, the common factor is y.

**Page 13 Exercise 4 Answer**

Given the polynomial

12x − x^{2}

Here it is asked to find the common factor.

Here we can see that x is common to both terms.

That is

12x − x^{2} = x(12−x)

Hence the common factor is x.

Therefore, the common factor is x.

**Page 13 Exercise 5 Answer**

Given the polynomial

3x^{3} + 2x^{2}

Here it is asked to find the common factor.

Here we can see that x^{2} is common to both terms.

That is

3x^{3} + 2x^{2} = x^{2}(3x+2)

Hence the common factor is x^{2}.

Therefore, the common factor is x^{2}.

**Page 13 Exercise 6 Answer**

Given the polynomial

x^{4} − 5x^{2}

Here it is asked to find the common factor.

Here we can see that x^{2} is common to both terms.

That is

x^{4} − 5x^{2} = x^{2}(x^{2}−5)

Hence the common factor is x^{2}.

Therefore, the common factor is x^{2}.

**Page 13 Exercise 7 Answer**

Given the polynomial

a^{3} + 5a^{2} + 3a

Here it is asked to find the common factor.

Here we can see that a is common to both terms.

That is

a^{3} + 5a^{2} + 3a = a(a^{2}+5a+3)

Hence the common factor is a.

Therefore, the common factor is a.

**Page 8 Exercise 13 Answer**

Given the polynomial

2x^{3} + x^{2} − 8x

Here it is asked to find the common factor.

Here we can see that x is common to both terms.

That is

2x^{3} + x^{2} − 8x = x(2x^{2}+x−8)

Hence the common factor is x.

Therefore, the common factor is x.

**Page 13 Exercise 9 Answer**

Given the polynomial

ab + 2b + b^{2}

Here it is asked to find the common factor.

Here we can see that b is common to both terms.

That is

ab + 2b + b^{2} = b(a+2+b)

Hence the common factor is b.

Therefore, the common factor is b.

**Page 13 Exercise 10 Answer**

Given the polynomial

5x^{2}y + xy + 7y

Here it is asked to find the common factor.

Here we can see that y is common to both terms.

That is

5x^{2}y + xy + 7y = y(5x^{2}+x+7)

Hence the common factor is y.

Therefore, the common factor is y.

**Page 13 Exercise 11 Answer**

Given the polynomial

x^{3} − 4x^{2} + x

Here it is asked to find the common factor.

Here we can see that x is common to both terms.

That is

x^{3} − 4x^{2} + x = x(x^{2}−4x+1)

Hence the common factor is x.

Therefore, the common factor is x.

**Page 13 Exercise 12 Answer**

Given the polynomial

ab + b^{2} + 2bc

Here it is asked to find the common factor.

Here we can see that b is common to both terms.

That is

ab + b^{2} + 2bc = b(a+b+2c)

Hence the common factor is b.

Therefore, the common factor is b.

**Page 14 Exercise 1 Answer**

Given the polynomial

5x^{2} + 10x

Here it is asked to find the common factor.

Here we can see that 5x is common to both terms.

That is

5x^{2} + 10x = 5x(x+2)

Hence the common factor is 5x.

Therefore, the common factor is 5x and can be written as 5x(x+2).

**Page 14 Exercise 2 Answer**

Given the polynomial

8a^{2} − 2a

Here it is asked to find the common factor.

Here we can see that 2a is common to both terms.

That is

8a^{2} − 2a = 2a(4a−1)

Hence the common factor is 2a.

Therefore, the common factor is 2a and can be written as 2a(4a−1).

**Page 14 Exercise 3 Answer**

Given the polynomial

4y^{5} + 3y^{3}

Here it is asked to find the common factor.

Here we can see that y^{3} is common to both terms.

That is

4y^{5} + 3y^{3} = y^{3}(4y^{2}+3)

Hence the common factor is y^{3}.

Therefore, the common factor is y^{3} and can be written as y^{3}(4y^{2}+3).

**Page 14 Exercise 4 Answer**

Given the polynomial

x^{2}y^{2} + x^{3}y

Here it is asked to find the common factor.

Here we can see that x^{2}y is common to both terms.

That is

x^{2}y^{2} + x^{3}y = x^{2}y(y+x)

Hence the common factor is x^{2}y.

Therefore, the common factor is x^{2}y and can be written as x^{2}y(y+x).

**Page 14 Exercise 5 Answer**

Given the polynomial

x^{2}y^{2} + xy

Here it is asked to find the common factor.

Here we can see that xy is common to both terms.

That is

x^{2}y^{2} + xy = xy(xy+1)

Hence the common factor is xy.

Therefore, the common factor is xy and can be written as xy(xy+1).

**Page 14 Exercise 6 Answer**

Given the polynomial

3x + 12x^{2}

Here it is asked to find the common factor.

Here we can see that 3x is common to both terms.

That is

3x + 12x^{2} = 3x(1+4x)

Hence the common factor is 3x.

Therefore, the common factor is 3x and can be written as 3x(1+4x).

**Page 14 Exercise 7 Answer**

Given the polynomial

6x^{3} − 9x^{2}

Here it is asked to find the common factor.

Here we can see that 3x^{2} is common to both terms.

That is

6x^{3} − 9x^{2} = 3x^{2}(2x−3)

Hence the common factor is 3x^{2}.

Therefore, the common factor is 3x^{2} and can be written as 3x^{2}(2x−3).

**Page 14 Exercise 8 Answer**

Given the polynomial

4x^{2} + 6xy

Here it is asked to find the common factor.

Here we can see that 2x is common to both terms.

That is

4x^{2} + 6xy = 2x(2x+3y)

Hence the common factor is 2x.

Therefore, the common factor is 2x and can be written as 2x(2x+3y).

**Page 14 Exercise 10 Answer**

Given the polynomial

12a^{2} − 18ab

Here it is asked to find the common factor.

Here we can see that 3a is common to both terms.

That is

12a^{2} − 18ab = 3a(4a−6b)

Hence the common factor is 3a.

Therefore, the common factor is 3a and can be written as 3a(4a−6b).

**Page 14 Exercise 11 Answer**

Given: Polynomial -7x^{3} − 7x^{2}

To find: A factorized form of a given polynomial.

factorize each of the terms in the given polynomial expression

⇒ 7 × x × x × x − 7 × x × x

As both of the terms have 7 × x × x common. So, from distributive law relation, we can rewrite it as

⇒ 7 × x × x(x−1)

⇒ 7x^{2}(x−1)

Hence, the final factorized form of the given polynomial is 7x^{2}(x−1)

**Page 14 Exercise 12 Answer**

Given: Polynomial -25x + 30xy

To find: A factorized form of a given polynomial.

Factorize each of the terms in the given polynomial expression

⇒ 5 × 5 × x + 2 × 3 × 5 × x × y

As both of the terms have a common 5 × x. So, from distributive law relation, we can rewrite it as

⇒ 5 × x(5+2×3×y)

⇒ 5x(5+6y)

Hence, the final factorized form of the given polynomial is 5x(5+6y)

**Page 14 Exercise 13 Answer**

Given: Polynomial -x^{6} + x^{5} − x^{4}

To find: A factorized form of a given polynomial.

Factorize each of the terms in the given polynomial expression

⇒ x × x × x × x × x × x + x × x × x × x × x − x × x × x × x

As both of the terms have common x × x × x × x. So, from distributive law relation, we can rewrite it as

⇒ x × x × x × x(x×x+x−1)

⇒ x^{4}(x^{2}+x−1)

Hence, the final factorized form of the given polynomial is x^{4}(x^{2}+x−1)

**Page 14 Exercise 15 Answer**

Given: Polynomial -5a^{3} + 3a^{4} + 6a^{3}

To find: A factorized form of a given polynomial.

Two of the terms have variable with the same power, so the like terms will get added directly,

⇒ 11a^{3} + 3a^{4}

Factorise each of the terms in the given polynomial expression

⇒ 11 × a × a × a + 3 × a × a × a × a

As, both of the terms have common a × a × a. So, from distributive law relation we can rewrite it as

⇒ a × a × a(11+3×a)

⇒ a^{3}(11+3a)

Hence, the final factorized form of the given polynomial is a^{3}(11+3a)

**Page 14 Exercise 16 Answer**

Given: Polynomial -6x^{2}y − xy^{2} + 2x^{2}y^{2}

To find: A factorized form of a given polynomial.

Factorise each of the terms in the given polynomial expression

⇒ 2 × 3 × x × x × y − x × y× y + 2 × x × x × y × y

As both of the terms have common x × y. So, from distributive law relation, we can rewrite it as

⇒ x × y(2×3×x−y+2×x×y)

⇒ xy(6x−y+2xy)

Hence, the final factorized form of the given polynomial is xy(6x−y+2xy)

**Page 14 Exercise 17 Answer**

Given: Polynomial -x^{3}y + x^{2}y + x^{2}y^{2}

To find: A factorized form of a given polynomial.

Factorise each of the terms in the given polynomial expression

⇒ x × x × x × y + x × x × y + x × x × y × y

As both of the terms have common x × x × y. So, from distributive law relation, we can rewrite it as

⇒ x × x × y(x+1+y)

⇒ x^{2}y(x+y+1)

Hence, the final factorized form of the given polynomial is x^{2}y(x+y+1)

**Page 14 Exercise 18 Answer**

Given: Polynomial -a^{3}b^{3} + a^{2}b^{2} + ab

To find: A factorized form of a given polynomial.

Factorise each of the terms in the given polynomial expression

⇒ a × a × a × b × b × b + a × a × b × b + a × b

As both of the terms have common a × b. So, from distributive law relation, we can rewrite it as

⇒ a × b(a×a×b×b+a×b+1)

⇒ ab(a^{2}b^{2}+ab+1)

Hence, the final factorized form of the given polynomial is ab(a^{2}b^{2}+ab+1)

**Page 14 Exercise 19 Answer**

Given: Polynomial -12a^{3} − 9a^{2} − 6a

To find: A factorized form of a given polynomial.

Factorise each of the terms in the given polynomial expression

⇒ 2 × 2 × 3 × a × a × a − 3 × 3 × a × a − 2 × 3 × a

As, both of the terms have a common 3 × a. So, from distributive law relation we can rewrite it as

⇒ 3 × a(2×2×a×a−3×a−2)

⇒ 3a(4a^{2}−3a−2)

Hence, the final factorized form of the given polynomial is 3a(4a^{2}−3a−2)

**Page 14 Exercise 23 Answer**

Given the polynomial to factorize: 63x^{4} + 81x^{3} − 72x^{2}

Now:

We have to factorize this polynomial by factoring out the biggest terms out of the polynomial.

Given the polynomial: 63×4+21×3-72×2

factorizing the expression

63x^{4} + 81x^{3} − 72x^{2}=9x^{2}(7x^{2}+9x-8)

Now see that:

⇒ \(\begin{aligned}

& \frac{63 x^4}{9 x^2}=7 x^2 \\

& \frac{81 x^3}{9 x^2}=9 x \\

& \frac{72 x^2}{9 x^2}=8

\end{aligned}\)

Here 9×2 is the biggest single term that can be taken out as a factor as it is the greatest term that is a factor of 63x^{4},81x^{3},72x^{2}

The polynomial after factoring out the biggest term is as follows: 63x^{4} + 81x^{3} − 72x^{2} = 9x^{2}(7x^{2}+9x−8)

**Page 14 Exercise 24 Answer**

Given the polynomial to factorize: 60a^{2} + 30ab − 90ac

Now:

We have to factorize this polynomial by factoring out the biggest terms out of the polynomial.

Given the polynomial 60a^{2}+30ab-90ac

factorizing the expression:

60a^{2}+30ab-90ac=30a(2a+b-3c)

Now see that:

⇒ \(\begin{aligned}

\frac{60 a^2}{30 a} & =2 a \\

\frac{30 a b}{30 a} & =b \\

\frac{90 a c}{30 a} & =3 c

\end{aligned}\)

Here 30a is the biggest single term that can be taken out as a factor as it is the greatest term that is a factor of 60a^{2},30ab,90ac

The polynomial after factoring out the biggest term is as follows: 60a^{2} + 30ab − 90ac = 30a(2a+b−3c)