Question

Expand and simplify each polynomial (Answer both problems) 1. (a^2-3a)^2 2. (1/2x^3 +6x)^2

Answer 1

The simplified form of polynomials are:

1.  (a² - 3a)² = 3a²

2. (1/2x³ + 6x)² = -12x²

Given,

We have to expand the given polynomials and simplify them.

1. The polynomial: (a² - 3a)²

Here,

The polynomial is in the form of (a - b)²

So,

(a - b)² = a² - 2ab + b²

Then,

(a² - 3a)² = (a²)² - (2 × a² × 3a) + (3a)²

(a² - 3a)² = a⁴ - 6a³ + 9a²

(a² - 3a)² = a²(a² - 6a + 9)

Here,

(a² - 6a + 9) is in quadratic form. Solve using quadratic formula.

a = 1, b = -6 and c = 9

$\frac{-b(+-)\sqrt{b^{2} -4ac} }{2a}$ = $\frac{6(+-)\sqrt{(-6)^{2}-4(1)(9) } }{2(1)}$ = $\frac{6(+-)\sqrt{36 - 36} }{2}$ = $\frac{6(+-)\sqrt{0} }{2}$ = (6±0) / 2

= 6/2 = 3

Then,

(a² - 3a)² = a²(a² - 6a + 9)

Here,

(a² - 6a + 9) = 3

So,

(a² - 3a)² = 3a²

2. The polynomial: (1/2x³ + 6x)²

Here, the polynomial is in the form (a + b)²

(a + b)² = a² + 2ab + b²

Here,

(1/2x³ + 6x)² = (1/2x³)² + (2 × 1/2x³ × 6x) + (6x)²

(1/2x³ + 6x)² = 1/4x⁶ + 6x⁴ + 36x²

(1/2x³ + 6x)² = x²(1/4x⁴ + 6x² + 36)

a = 1/4, b = 6 and c = 36

Quadratic formula: $\frac{-b(+-)\sqrt{b^{2} -4ac} }{2a}$

= $\frac{-6(+-)\sqrt{(-6)^{2} -4(1/4)(36)} }{2(1/4)}$ = $\frac{-6(+-)\sqrt{36-36} }{1/2}$ = (-6±0) / (1/2) = -6 × 2 = -12

Therefore,

(1/2x³ + 6x)² = x²(1/4x⁴ + 6x² + 36)

Here,

(1/4x⁴ + 6x² + 36) = -12

Then,

(1/2x³ + 6x)² = -12x²

That is,

The simplified form of polynomials are:

1.  (a² - 3a)² = 3a²

2. (1/2x³ + 6x)² = -12x²

Learn more about simplification of polynomials here:

brainly.com/question/355399

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