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Polynomials — Maths STD 9 — Question

Maharashtra BoardEnglish MediumSTD 9MathsPolynomials4 Marks
Question
By using factor theorem in the following examples, determine whether $q(x)$ is a factor of $p(x)$ or not.
i. $p(x)=x^3-x^2-x-1 ; q(x)=x-1$
ii. $p(x)=2 x^3-x^2-45 ; q(x)=x-3$
✓

Answer

i. $p(x)$ =$ x^3 – x^2 – x – 1$
Divisor =$q(x)$ $= x – 1$
∴ take $x = 1$
Remainder = p(1)
$p(x) = x^3 – x^2 – x – 1$
$\therefore P(1) = (1)^3 – (1)^2 – 1 – 1$
$= 1 – 1 – 1 – 1$
$= -2 \neq 0$
∴ By factor theorem, $x – 1$ is not a factor of$ x^3 – x^2 – x – 1$.ii. $p(x) = 2x^3 – x – 45$
Divisor $= q(x) = x – 3$
take $x = 3$
Remainder =$ p(3)$
$p(x) = 2x^3 – x^2 – 45$
$P(3) = 2(3)^3 – (3)^2 – 45$
$= 2(27) – 9 – 45$
$= 54 – 9 – 45$
$= 0$
$\therefore$ By factor theorem, $x – 3$ is a factor of $2x^3 – x^2 – 45.$

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