Question

Use factor theorem to determine that x- 3 is factor of x cube - 4x square + x + 6

Answer 1

Answer :

(x - 3) is the factor of x³ - 4x² + x + 6.

Explanation :

Let f(x) = x³ - 4x² + x + 6

$when\:f(x)\div(x-3)$

By factor theorem when f(x) / (x - a) i, f(a) = 0

To know the remainder find the zero of x - 3

To find zero equate x - 3 to 0

x - 3 = 0

x = 3

So, f(3) is the remainder

f(3) = (3)³ - 4(3)² + 3 + 6

= 27 - 4(9) + 9

= 36 - 36

f(3) = 0

So, f(3) = 0

By factor therem (x - 3) is the factor of x³ - 4x² + x + 6.

Extra info :

Zero of a polynomial : We say that a zero of a polynomial p(x) is the value of x, which p(x) is equal to zero. this value is also called a root of the polynomial p(x).

Polynomial: An algebraic expression in which the variables involved have only non-zero negative integral powers is called a polynomial.

Cubic polynomial : A polynomial of degree 3 is called the cubic polynomial.

• If the variable in a polynomial is x, we made not the polynomial by p(x), q(x) or r(z) etc.

e.g. p(x) = 3x² + 2x + 1

q(x) = x³ - 5x² + x - 7

r(y) = y³ - 1

r(y) = y³ - 1t(z) = z² + 5z + 3

Factor theorem : If x - a is a factor of the polynomiap p(x), then p(a) = 0. Also if p(a) = 0 then (x - a) is a factor of p(x).