Question

$\sf \: If\: {x}^{2} -4\: is \: a \: factor \: of \: polynomial \: {x}^{3} + {x}^{2} - 4x - 4, \: then \: its \: factors \: are \: ?$
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Answer 1

Given polynomial is p(x) = x³ + x² – 4x – 4

x – 2 is its factor, if p(2) = 0

p(2) = (2)3 + (2)2 – 4(2) – 4 = 8 + 4 – 8 – 4 = 0

Thus, x – 2 is a factor of p(x).

Now, x³ + x²– 4x + 4 = x²(x + 1) – 4(x + 1)

= (x + 1) (x² – 4)

= (x + 1) (x + 2) (x – 2)

Hence, the required factors are (x + 1), (x + 2) and

(x – 2).

Answer 2

Question

If x² - 4 is a factor of polynomial x³ + x² - 4x - 4, then its factors are :-

Answer

The required factors of given polynomial, take

$\tt{f(x) = (x-2)(x+2)(x+1)}$

Concept

Given that, x² - 4 is a factor of the given polynomial f(x) = x³ + x² - 4x - 4. Let's write the factor in the simplified form first. x² - 4 = 0 ; this implies that (x+2)(x-2) is a factor of f(x) by Remainder and Factor Theorem's principle but it's not the only factor to find the other factors too we apply the basic sense that if a times b gives the product c then along with a , b will also be the factor of c. Similarly, if (x+2)(x-2) is a factor of f(x) then Let's find the one more factor multiplied to which it bears the exact polynomial.

Calculation of third factor

$\begin{array}{c|c}& \rm \red{{x}^{2} - x - 2} \\ \hline\sf x + 2 & \sf \cancel{x}^{3} +{x}^{2} - 4x - 4 \\ &( - ) \sf \: \cancel {x}^{3} + 2 {x}^{2}\\& - \: - \\ \hline& \sf \pink{ \cancel{- x ^{2}} - 4x}\\& (- )\pink{ \cancel{- \sf {x}^{2}} - 2x } \\& ( + ) ( + ) \\ \hline & \orange{\sf \cancel{- 2x} - \cancel 4 }\\& ( - )\orange{ \sf \cancel{- 2x} - \cancel4 }\\& - \: - \\ \hline &0 \\ \hline\end{array}$

Thus, other then (x+2) we got one more factor i.e x² - x - 2 = (x-2)(x+1) since we know (x-2) as a factor what new factor we get is (x+1) therefore

Factor of f(x) = (x+2)(x-2)(x+1)

$\underline{\rule{190pt}{2pt}}$

Remainder Theorem

Remainder Theorem states that if f(x) a polynomial is divided by x-a then the remainder would be f(a).

Factor Theorem

Factor Theorem states that if f(a) is a remainder of polynomial f(x) which bears a remainder 0 then this means that (x-a) is a factor of that polynomial.

Thankyou