Question

Consider the polynomials:

A(x) = (3-x)2² - (x-3) (7x+4)-18+2x²
B(x) = (3x + 2)² - (x - 1)²
1) Develop and reduce A (x).
2) Factorize A (x) and B (x).
3) Solve the equation A (x)= B(x)
4) Prove that B (X) -3 = 2x(4x+7) , then solve the equation B(x) =3

Answer 1

1) -5x² + 13x + 6

2) B(x) =(4x + 1 )(2x + 3)

A(x) = (5x +2)(3-x)

Explanation:

1) Expand the equation by using distributive property. Then simplify by combining the like terms.

A(x) = (3 -x)2² - (x -3)(7x+ 4) - 18 + 2x²

= (3 - x)4 - [x*7x + x*4 - 3*7x - 3*4] - 18 + 2x²

= 3*4 - x*4 - [7x² + 4x - 21x - 12] - 18 + 2x²

= 12 - 4x - 7x² - 4x + 21x + 12 - 18 + 2x²

= 2x² - 7x² -4x - 4x + 21x + 12 + 12 - 18

= -5x² + 13x + 6

B(x) = (3x + 2)² - (x - 1)²

Identities:

(a + b)² = a² + 2ab + b² ; Here a =3x & b = 2

(m -n)² = m² - 2mn + n² ; Here m = x & n = 1

2) B(x) = (3x)² + 2*3x*2 + 2² - [x² - 2*x*1 + 1]

= 9x² + 12x + 4 - [x² - 2x + 1]

= 9x² + 12x + 4 - x²+ 2x - 1

= 9x² - x² + 12x + 2x + 4 - 1

= 8x² + 14x +3

Product = 8*3 = 24

Sum = 14

Factors = 2 , 12 {2*12 = 24 & 2 +12= 14}

= 8x² + 2x + 12x + 3 {Rewrite the middle term}

= 2x(4x + 1) + 3(4x + 1)

= (4x + 1 )(2x + 3)

A(x) = - 5x² + 13x + 6

= -5x² + 15x -2x + 6

= 5x(-x +3) +2 (-x + 3)

= (5x +2)(3-x)

4) B(x) - 3 = 8x² + 14x + 3 - 3

= 8x² + 14x

= 2x *4x + 2x *7

=2x(4x + 7)

Hence proved.

B(x) = 3

8x² + 14x + 3 = 3

8x² + 14x + 3 - 3 = 0

8x² + 14x = 0

2x(4x + 7) = 0

2x = 0 or 4x + 7 = 0

x = 0 or 4x = -7

`\sf x = (-7)/(4)`

x = 0 or
`\sf (-7)/(4)`