Question

How do you factor given that \[f\left( { - 5} \right) = 0\]and \[f\left( x \right) = 4{x^3} + 9{x^2} - 52x + 15?\]

Answer 1

Hint: In the given question, we have been asked to find factors of a quadratic equation with a single variable. Since in the equation variable has degree 3, $ x $ will have three values. Factoring of linear expression is basically representing it in the form $ (x - \alpha )(x - \beta )(x - \gamma ) $ . In order to proceed with the following question we need to use a long division method.

Complete step by step solution:
We are given,
\[f\left( x \right) = 4{x^3} + 9{x^2} - 52x + 15?\]
Since it is given the expression becomes\[\;0\]when\[x = - 5\], we can state that \[x + 5\]is a factor of the given expression.
We can find other factors by dividing the expression by its factor.
\[
  x + 5\mathop{\left){\vphantom{1{4{x^3} + 9{x^2} - 52x + 15}}}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{{4{x^3} + 9{x^2} - 52x + 15}}}}
\limits^{\displaystyle \,\,\, { - 4{x^2} + 11x - 3}} \\
  \;\;\;\,\,\,\,\,\, - \underline {4{x^3} - 20{x^2}} \\
  \;\,\,\,\,\;\;\;\;\;\;0\;\;\;\; - 11{x^2} - 52x \\
  \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\underline { + 11{x^2} + 55x} \\
  \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;0\;\;\;\;\; + 3x + 15 \\
  \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; - \underline {3x + 15} \\
  \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,\underline {\;\;\;00\;\;} \\
 \]
 $ \Rightarrow (x + 5)( - 4{x^2} + 11x - 3) $
We can rewrite the above equation by making the \[{x^2}\]term positive.
 $ \Rightarrow (x + 5)(4{x^2} - 11x + 3) $
This equation cannot be further factorized; \[x\]has only one rational value.
So, this is the required solution. $ $
So, the correct answer is “ $ (x + 5)(4{x^2} - 11x + 3) $ ”.

Note: Before solving any question of quadratic equation, ensure that the equation is of the form $ a{x^2} + bx + c = 0 $ , and if it is not then convert it in this form, where $ a,b,c \in R $ and $ a \ne 0 $ . Polynomial Long Division method is basically dividing one polynomial expression by another of the same degree or less. Degree is basically the highest power of any algebraic expression. In this method we divide, multiply, subtract, bring down and repeat until we get a remainder. This method can also be checked by the formula: Dividend= Divisor x Quotient + Remainder.
For dividing polynomials, there as three cases:
Division of a monomial expression by another monomial expression
Division of a polynomial expression by monomial expression
Division of a polynomial expression by binomial expression
Division of a polynomial expression by another polynomial expression.