10 Factor the following polynomials a x2 3x 4 b 2x2 3x 27 c

10. Factor the following polynomials: a. x2 - 3x -4 b. 2x2 +3x -27 c. 2x2 +x-36 d. x2-9x 14 e. x2-2x +5 4x2-4x + 3 g. 3x512x12x3 11. Factor the following polynomials a. 12x3- 21x2 + 28x-49 b. 25x3 5x2 + 30x 6 c. 12x3 2x2-30x 5 e. x2 + x +1 f. 8x3 - 64x2 + x- 8 8. 25x2 - 36y2

Solution

!0 a. x² - 3x - 4 = x² -4x + x - 4 = x (x - 4) +1(x -4) = (x - 4) (x + 1)

b. 2x² + 3x - 27 = 2x² + 9x - 6x -27 = 2x (x+9/2) - 6 (x + 9/2) = ( x + 9/2) ( 2x -6) = 2 (x + 9/2)( x- 3) = (2x + 9)(x - 3)

c. 2x² + x - 36 = 2x² + 9x -8x - 36 = x ( 2x + 9) -4( 2x + 9) = ( 2x + 9)( x -4)

d. x² -9x + 14 = x² - 2x - 7x + 14 = x(x -2) + 7 (x - 2) = ( x - 2) ( x + 7)

NOTE: Whereever, we find it difficult, please remember that the roots pf the polynomial ax² + bx + c are [-b±(b² – 4ac)]/2a

1.

A monomial is the product of non-negative integer powers of variables. Consequently, a monomial has no variable in its denominator. It has one term. (mono implies one) e.g. 5x, 3x², - xy etc.

A binomial is the sum of two monomials. It has two unlike terms.  
(bi implies two) e.g. 2x + 1,    5x² - 9x,     7x + 2y etc.

A trinomial is the sum of three monomials. It has three unlike terms.   (tri implies three) e.g.
3x² + 2x + 9,      2x² - 5x + 10,       x + 3y + 2 etc.

2. a. Degree 2, Leading coefficient 4

b. Degree 3, Leading coefficient - 2

c. Degree 1, Leading coefficient 1

3. a. 6y -9y = -3y

b. 8 - 2x³ + 5x + 21x²   There is nothing to combine

c. -1 + x - x² + x³    There is nothing to combine

4. a. (4x² - 2x + 1) + (3x² -5x + 6) = 4x² + 3x² -2x - 5x + 1 + 6 = 7x² -7x + 7

b. (7x³ - 5x -2x² + 1) + ( 7x - 2 + 2x²) = 7x³ -2x² + 2x² -5x + 7x + 1 - 2 = 7x³ + 2x -1

c. (4x⁶ - 7x⁴ - 9x² + 1) + ( -7x⁶ + 7x⁴-9x² -11) = 4x⁶ - 7x⁶ - 7x⁴+ 7x⁴- 9x² -9x² + 1 -11 = -3x⁶ - 18x² - 10

6. Let A = 2x² - 3x + 1

When x = 3, A = 2 (3)² - 3 (2) + 1 = 18 -6 + 1 = 13

When x = -2, A = 2 ( -2)² - 3 (-2) + 1 = 8 + 6 + 1 = 15

When x = y, A = 2y² -3y + 1

7 a. True ( except when the result is 0)

b. True

c. False

8. a -6x² + 12x = - 6x ( x -2)

b. 9x³ - 18 x⁴ = -18x⁴ + 9x³ = -9x³ ( 2x -1)

c.8xy³ + 4xy² = 4xy²( 2y + 1)