Determine which of the following polynomials has $ (x+1) $ a factor:
(i) $ x^{3}+x^{2}+x+1 $
(ii) $ x^{4}+x^{3}+x^{2}+x+1 $
Given :
The given term is $(x + 1)$.
The given polynomials are
(i) \( x^{3}+x^{2}+x+1 \)
(ii) \( x^{4}+x^{3}+x^{2}+x+1 \)
To find :
We have to check whether the given polynomials have $(x + 1)$ as a factor.
Solution :
According to factor theorem,
If $(x+1)$ is a factor of given polynomial $P(x)$, then at $x= -1$, $p(x)=0$.
(i) $x^{3}+x^{2}+x+1$
Let $p(x)= x^{3}+x^{2}+x+1$
Substituting $x= -1$
$p(−1)=(−1)^3+(−1)^2+(−1)+1 =−1+1−1+1=0$
Hence, by factor theorem, $x+1$ is a factor of $x^{3}+x^{2}+x+1$.
(ii) $x^{4}+x^{3}+x^{2}+x+1 $
Let $p(x)=x^{4}+x^{3}+x^{2}+x+1 $
Substituting $x= -1$
$p(−1)=(−1)^4+(-1)^3+(−1)^2+(−1)+1 =1−1+1−1+1=1$
Hence, by factor theorem, $x+1$ is not a factor of $x^{4}+x^{3}+x^{2}+x+1$.
Related Articles
- Determine which of the following polynomials has \( (x+1) \) a factor:(i) \( x^{3}+x^{2}+x+1 \)(ii) \( x^{4}+x^{3}+x^{2}+x+1 \)(iii) \( x^{4}+3 x^{3}+3 x^{2}+x+1 \)(iv) \( x^{3}-x^{2}-(2+\sqrt{2}) x+\sqrt{2} \)
- Determine if the following polynomials has $(x+1)$ a factor: $x^{3}+x^{2}+x+1$
- Check whether the following are quadratic equations:(i) \( (x+1)^{2}=2(x-3) \)(ii) \( x^{2}-2 x=(-2)(3-x) \)(iii) \( (x-2)(x+1)=(x-1)(x+3) \)(iv) \( (x-3)(2 x+1)=x(x+5) \)(v) \( (2 x-1)(x-3)=(x+5)(x-1) \)(vi) \( x^{2}+3 x+1=(x-2)^{2} \)(vii) \( (x+2)^{3}=2 x\left(x^{2}-1\right) \)(viii) \( x^{3}-4 x^{2}-x+1=(x-2)^{3} \)
- Use the Factor Theorem to determine whether \( g(x) \) is a factor of \( p(x) \) in each of the following cases:(i) \( p(x)=2 x^{3}+x^{2}-2 x-1, g(x)=x+1 \)(ii) \( p(x)=x^{3}+3 x^{2}+3 x+1, g(x)=x+2 \)(iii) \( p(x)=x^{3}-4 x^{2}+x+6, g(x)=x-3 \)
- If \( x+\frac{1}{x}=3 \), calculate \( x^{2}+\frac{1}{x^{2}}, x^{3}+\frac{1}{x^{3}} \) and \( x^{4}+\frac{1}{x^{4}} \).
- Simplify the following :$( 3 x^2 + 5 x - 7 ) (x-1) - ( x^2 - 2 x + 3 ) (x + 4)$
- Solve for $x$:$\frac{x-1}{x-2}+\frac{x-3}{x-4}=3\frac{1}{3}, x≠2, 4$
- If \( x^{4}+\frac{1}{x^{4}}=194 \), find \( x^{3}+\frac{1}{x^{3}}, x^{2}+\frac{1}{x^{2}} \) and \( x+\frac{1}{x} \)
- Verify whether the following are zeroes of the polynomial, indicated against them.(i) \( p(x)=3 x+1, x=-\frac{1}{3} \)(ii) \( p(x)=5 x-\pi, x=\frac{4}{5} \)(iii) \( p(x)=x^{2}-1, x=1,-1 \)(iv) \( p(x)=(x+1)(x-2), x=-1,2 \)(v) \( p(x)=x^{2}, x=0 \)(vi) \( p(x)=l x+m, x=-\frac{m}{l} \)(vii) \( p(x)=3 x^{2}-1, x=-\frac{1}{\sqrt{3}}, \frac{2}{\sqrt{3}} \)(viii) \( p(x)=2 x+1, x=\frac{1}{2} \)
- Identify polynomials in the following:\( h(x)=x^{4}-x^{\frac{3}{2}}+x-1 \)
- Identify polynomials in the following:\( g(x)=2 x^{3}-3 x^{2}+\sqrt{x}-1 \)
- Simplify:$(x^3 - 2x^2 + 3x - 4) (x - 1) - (2x - 3) (x^2 - x + 1)$
- Find the remainder when \( x^{3}+3 x^{2}+3 x+1 \) is divided by(i) \( x+1 \)(ii) \( x-\frac{1}{2} \)(iii) \( x \)(iv) \( x+\pi \)(v) \( 5+2 x \)
- Simplify each of the following products:\( (x^{3}-3 x^{2}-x)(x^{2}-3 x+1) \)
- Factorise:(i) \( 12 x^{2}-7 x+1 \)(ii) \( 2 x^{2}+7 x+3 \)(iii) \( 6 x^{2}+5 x-6 \)(iv) \( 3 x^{2}-x-4 \)
Kickstart Your Career
Get certified by completing the course
Get Started