Question
Using factor theorem, factorize the following polynomials:
$x^3 - 10x^2 - 53x - 42$
$x^3 - 10x^2 - 53x - 42$
✓
Answer
Let $f(x) = x^3 - 10x^2 - 53x - 42$ be the given polynomial.
Now, putting $x = -1$, we get
$f(-1) = (-1)^3 - 10(-1)^2 - 53(-1) - 42$
$= -1 - 10 + 53 - 42$
$= -53 + 53 = 0$
Therefore, $(x + 1)$ is a factor of polynomial$ f(x).$
Now,
$f(x) = x^2(x + 1) - 11x(x + 1) - 42(x + 1)$
$= (x + 1)(x^2 - 11x - 42)$
$= (x + 1)(x^2 - 14x + 3x - 42)$
$= (x + 1)(x + 3)(x - 14)$
Hence $(x + 1), (x + 3)$ and $(x - 14)$ are the factors of polynomial$ f(x).$
Now, putting $x = -1$, we get
$f(-1) = (-1)^3 - 10(-1)^2 - 53(-1) - 42$
$= -1 - 10 + 53 - 42$
$= -53 + 53 = 0$
Therefore, $(x + 1)$ is a factor of polynomial$ f(x).$
Now,
$f(x) = x^2(x + 1) - 11x(x + 1) - 42(x + 1)$
$= (x + 1)(x^2 - 11x - 42)$
$= (x + 1)(x^2 - 14x + 3x - 42)$
$= (x + 1)(x + 3)(x - 14)$
Hence $(x + 1), (x + 3)$ and $(x - 14)$ are the factors of polynomial$ f(x).$
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