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### The polynomial x^3 - 2*x + a, where x is a constant is denoted by p(x). It is given that x+2 is a factor of p(x). Find a

Before solving this question, lets look at a second order polynomial.

Let the polynomial x^2 +5*x +4 be f(x)

Using factorisation, we can write f(x) as (x+1)(x+4)

If you were to expand (x+1)(x+4) you'd get x^2 +5*x +4.

To solve f(x) = 0, we can write (x+1)(x+4) = 0 and then set each one of the factors to zero.

x+1 = 0 --> x = - 1 and x+4 = 0 --> x = - 4

Check that: f(-1) = 0 and that f(-4) = 0

We can conclude that,

- setting the factors of a polynomial f(x) to zero gives the roots of the equation f(x) = 0

- plugging the roots (values of x) back in the polynomial expression will lead to f(x) equalling zero.

Coming back to the question, we know that the factor of polynomial p(x) is (x+2).

Setting x+ 2 equal to zero gives x = - 2

This is a one of the three roots/solutions of the equation p(x) = 0 and plugging it into the polynomial expression should give zero,

i.e, p(-2) = 0

p(-2) = -2^3 - 2*(-2) + a = 0

-8 + 4 +a = 0

-4 + a = 0

a = 4

- setting factor of p(x) to zero gives root of the equation p(x)=0

-Plugging root into p(x) expression will satisfy p(x)=0

1 year ago

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