Question

From the following polynomials, decide which has (x + 1) or (x - 1) as a factor.
(I) p(x) = 3x³ - 7x² + 5x - 1
(2) p(x) = 21x³ + 16x² + 4x + 9
(3) p(x) = 2x⁴ - 3x³ + 4x² - 5x + 2
(4) p(x) = x³ + 13x² + 32x + 20

Answer 1

Answer:

From the following polynomials, decide which has (x + 1) or (x - 1) as a factor.

(I) p(x) = 3x³ - 7x² + 5x - 1

(2) p(x) = 21x³ + 16x² + 4x + 9

(3) p(x) = 2x⁴ - 3x³ + 4x² - 5x + 2 

(4) p(x) = x³ + 13x² + 32x + 20

1.

$p(x)=3x^3-7x^2+5 -1$

$\text{Sum of the coefficients=3-7+5-1=0}$

$\therefore\:\text{(x-1) is a factor of (x)}$

2.

$p(x)=21x^3+16x^2+4x+9$

$\text{Sum of the coefficients of even powers of x=16+9=25}$

$\text{Sum of the coefficients of odd powers of x=21+4=25}$

$\implies\:\text{Sum of the coefficients of even powers of x=Sum of the coefficients of odd powers of x}$

$\therefore\:\text{(x+1) is a factor of (x)}$

3.

$p(x)=2x^4-3x^3+4x^2-5x+2$

$\text{Sum of the coefficients=2-3+4-5+2=0}$

$\therefore\:\text{(x-1) is a factor of (x)}$

4.

$p(x)=x^3+13x^2+32x+20$

$\text{Sum of the coefficients of even powers of x=13+20=13}$

$\text{Sum of the coefficients of odd powers of x=1+32=33}$

$\implies\:\text{Sum of the coefficients of even powers of x=Sum of the coefficients of odd powers of x}$

$\therefore\:\text{(x+1) is a factor of (x)}$