Proving Polynomial Identities

a

Here is a parentheses which has been raised by

2 .

We can expand it, using the square of a binomial.

3 (x + 5)^{2} - 75

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

3 (x^{2} + 2 \cdot x \cdot 5 + 5^{2}) - 75

Simplify

Simplify power and product

3 (x^{2} + 10 x + 25) - 75

DistrDistribute

3

3 \cdot x^{2} + 3 \cdot 10 x + 3 \cdot 25 - 75

3 x^{2} + 30 x + 75 - 75

SubTermSubtract term

3 x^{2} + 30 x

b

Looking at the first term, we can use the difference of squares. First, let us rearrange the order of the parentheses.

(7 - x) (x + 7) + 49

CommutativePropMultCommutative Property of Multiplication

(7 + x) (7 - x) + 49

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

7^{2} - x^{2} + 49

Simplify

CalcPowCalculate power

49 - x^{2} + 49

AddTermsAdd terms

98 - x^{2}

c

Now we use both the difference of squares and the squared binomials.

(x - 4) (x + 4) - (x - 4)^{2}

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

x^{2} - 4^{2} - (x - 4)^{2}

Simplify

CalcPowCalculate power

x^{2} - 16 - (x - 4)^{2}

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

x^{2} - 16 - (x^{2} - 2 \cdot x \cdot 4 + 4^{2})

Simplify power and product

x^{2} - 16 - (x^{2} - 8 x + 16)

RemoveParSignsRemove parentheses and change signs

x^{2} - 16 - x^{2} + 8 x - 16

CommutativePropAddCommutative Property of Addition

x^{2} - x^{2} + 8 x - 16 - 16

SimpTermsSimplify terms

8 x - 32

Proving Polynomial Identities 1.13 - Solution