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This is one of our many quizzes on Factoring polynomials. Click on this to access more such competitive quizzes on Algebra.

Factoring polynomials Quiz 2

mathematics

 Find $({{x}^{2}}+4)+({{x}^{2}}+8)-({{x}^{2}}-8)-({{x}^{2}}+4)$

 Use addition and subtraction operation on polynomials.

$({{x}^{2}}+4)+({{x}^{2}}+8)-({{x}^{2}}-8)-({{x}^{2}}+4)  \\ \Rightarrow {{x}^{2}}+4+{{x}^{2}}+8-{{x}^{2}}-8-{{x}^{2}}+4  \\ \Rightarrow 8 $

 Find the degree of polynomial if $9{{x}^{123}}+65{{x}^{999}}-55{{x}^{12}}-{{88}^{125}}+7899{{x}^{0}}$

 Use the concept of degree of polynomial.

 Here, the highest power of $x$ is $999$, $ \\ $ So, the degree of polynomial is $999$.

 If $P \left( x \right)={{x}^{3}}-{{x}^{2}}+5$ then find the value of $p(5)-p(4)+p(3)-p(2)+p(1)$

 Use Substitution method and find answers by substituting values in given polynomials.

Here, $ p(5)-p(4)+p(3)-p(2)+p(1)$  $p(5)=125-25+5=105 \\   p(4)=64-16+5=53 \\   p(3)=27-9+5=23 \\   p(2)=8-4+5=9 \\   p(1)=1-1+5=5 \\ $ Thus,  $p(5)-p(4)+p(3)-p(2)+p(1)  \\ \Rightarrow 105-53+23-9+5  \\ \Rightarrow 71 $

 Which polynomial is to be multiplied to $5p+2q+3$ to get $5{{p}^{3}}+15p+2{{p}^{2}}q+6q+3{{p}^{2}}+6q+9$?

 Use multiplication and division of polynomials.

 $(5p+2q+3)A=5{{p}^{3}}+15p+2{{p}^{2}}q+6q+3{{p}^{2}}+6q+9 \\   A=\dfrac{5{{p}^{3}}+15p+2{{p}^{2}}q+6q+3{{p}^{2}}+6q+9}{5p+2q+3} \\   A={{p}^{2}}+3$

 If $p$ and $q$ are constants and ${{x}^{2}} +px +6=\left( x+2 \right)\left( x+q \right)$, what is the value of $p$?

 solve by using factorisation techniques and expansion.

 We first expand the RHS as ${{x}^{2}}+2x+qx+2q$ and factor out the $x$ terms to get ${{x}^{2}}+\left( 2+q \right)x+2q$. We then set this equal to the original LHS to get ${{x}^{2}}+px +6={{x}^{2}}+\left( 2+q \right)x+2q$, and then subtract ${{x}^{2}}$ from each side to get $px+6=\left( 2+q \right)x+2q$ Since the coefficients of the $x$ terms on both sides must be equal, and the constant terms on both sides must be equal, we find that $p=2+q$ and $6=2q$, so $q$ is equal to $3$ and $p$ is equal to $5$.