Show that x+a is a factor of the polynomial P(x)=(x+a)^4+(x+c)^2-(a-c)^2

Question

Use factor theorem

Answer 1

P(x)=(x+a)⁴+(x+c)²-(a-c)²

we are going to simplify (x+c)² - (a-c)² using the formula A² - B² = (A - B) (A + B)

P(x)= (x + a)⁴ + (x + c)²- (a - c)² = (x + a)⁴ + (x + c + a - c) (x + c - a + c) = (x + a)⁴ + (x + a) (x + 2c - a)

Now we are going to factor out (x + a)

P(x)= (x + a)⁴ + (x + c)²- (a - c)² = (x + a)⁴ + (x + c + a - c) (x + c - a + c) = (x + a)⁴ + (x + a) (x + 2c - a) =

(x + a) [ (x + a)³ + x + 2c - a ]

Alternatively we can use the factor theorem, substitute x = - a, and find P (-a)

P(-a)=(-a+a)⁴+(-a+c)²-(a-c)² = 0⁴ + (c-a)² - (a-c)² = 0

This is because

0⁴ = 0 and (c-a)² = (a-c)²

P (-a) = 0, therefore (-a) is a root of P, therefore (x - (-a)) = (x + a) is a factor of P(x)