Mix Examples - Polynomials Questions in English

(A) To find the value of $p(2)$,substitute $x = 2$ into the given polynomial $p(x) = x^{2} - 7x + 12$.
$p(2) = (2)^{2} - 7(2) + 12$
$p(2) = 4 - 14 + 12$
$p(2) = 16 - 14$
$p(2) = 2$
Therefore,the correct option is $A$.

(B) According to the Factor Theorem,if $p(a) = 0$ for a polynomial $p(x)$,then $(x - a)$ is a factor of $p(x)$.
Given that $p(7) = 0$,we substitute $a = 7$ into the theorem.
Therefore,$(x - 7)$ is a factor of $p(x)$.

(A) According to the Factor Theorem,if $p(a) = 0$ for a polynomial $p(x)$,then $(x - a)$ is a factor of $p(x)$.
Given that $p(3) = 0$,we substitute $a = 3$ into the theorem.
Therefore,$(x - 3)$ is a factor of $p(x)$.

(D) To find the remainder when $p(x) = 3x^3 - 6x^2 + 5x - 10$ is divided by $(x - 2)$,we use the Remainder Theorem.
According to the Remainder Theorem,if a polynomial $p(x)$ is divided by $(x - a)$,the remainder is $p(a)$.
Here,$a = 2$.
Substitute $x = 2$ into the polynomial $p(x)$:
$p(2) = 3(2)^3 - 6(2)^2 + 5(2) - 10$
$p(2) = 3(8) - 6(4) + 10 - 10$
$p(2) = 24 - 24 + 10 - 10$
$p(2) = 0$
Therefore,the remainder is $0$.

(A) To determine if $(x-1)$ is a factor of the polynomial $p(x) = 3x^2 + 7x - 10$,we use the Factor Theorem.
According to the Factor Theorem,$(x-a)$ is a factor of $p(x)$ if $p(a) = 0$.
Here,$a = 1$.
Substitute $x = 1$ into the polynomial:
$p(1) = 3(1)^2 + 7(1) - 10$
$p(1) = 3(1) + 7 - 10$
$p(1) = 3 + 7 - 10$
$p(1) = 10 - 10 = 0$.
Since $p(1) = 0$,$(x-1)$ is indeed a factor of $3x^2 + 7x - 10$.

(B) To find $p(-2)$,substitute $x = -2$ into the given polynomial $p(x) = 5x^2 - 11x + 3$.
$p(-2) = 5(-2)^2 - 11(-2) + 3$
$p(-2) = 5(4) + 22 + 3$
$p(-2) = 20 + 22 + 3$
$p(-2) = 45$
Therefore,the value of $p(-2)$ is $45$.

(B) The given expression is $49x^2 - 121$.
This can be written in the form of $a^2 - b^2$ as follows:
$49x^2 - 121 = (7x)^2 - (11)^2$.
Using the algebraic identity $a^2 - b^2 = (a + b)(a - b)$,where $a = 7x$ and $b = 11$,we get:
$(7x)^2 - (11)^2 = (7x + 11)(7x - 11)$.
Therefore,the factors are $(7x + 11)(7x - 11)$.

(D) Given the equation: $(5x - 3)^2 = 25x^2 + kx + 9$.
We use the algebraic identity $(a - b)^2 = a^2 - 2ab + b^2$.
Here,$a = 5x$ and $b = 3$.
Expanding the left side: $(5x - 3)^2 = (5x)^2 - 2(5x)(3) + (3)^2$.
$(5x - 3)^2 = 25x^2 - 30x + 9$.
Comparing this with the given expression $25x^2 + kx + 9$,we get:
$25x^2 - 30x + 9 = 25x^2 + kx + 9$.
By comparing the coefficients of $x$,we find $k = -30$.

(A) To find the value of $k$,we expand the left side of the equation:
$(2x + 3)(3x - 1) = 2x(3x) + 2x(-1) + 3(3x) + 3(-1)$
$= 6x^2 - 2x + 9x - 3$
$= 6x^2 + 7x - 3$
Comparing this with the given equation $6x^2 + kx - 3$,we can see that the coefficient of $x$ is $k$ on the right side and $7$ on the expanded left side.
Therefore,$k = 7$.

(B) To find the remainder when $16x^2 - 24x + 9$ is divided by $4x - 3$,we can use the Remainder Theorem or perform polynomial long division.
Method $1$: Factorization
Notice that $16x^2 - 24x + 9$ is a perfect square trinomial.
$16x^2 - 24x + 9 = (4x)^2 - 2(4x)(3) + (3)^2 = (4x - 3)^2$.
Since $(4x - 3)^2$ is exactly divisible by $(4x - 3)$,the remainder is $0$.
Method $2$: Remainder Theorem
Set the divisor equal to zero: $4x - 3 = 0 \implies x = 3/4$.
Substitute $x = 3/4$ into the polynomial $P(x) = 16x^2 - 24x + 9$:
$P(3/4) = 16(3/4)^2 - 24(3/4) + 9$
$P(3/4) = 16(9/16) - 18 + 9$
$P(3/4) = 9 - 18 + 9 = 0$.
Thus,the remainder is $0$.

(A) To determine if $(x+1)$ is a factor of the polynomial $p(x) = 4x^3 + 7x^2 - 2x - 5$,we use the Factor Theorem.
According to the Factor Theorem,$(x-a)$ is a factor of $p(x)$ if $p(a) = 0$.
Here,we set $x+1 = 0$,which gives $x = -1$.
Now,we calculate $p(-1)$:
$p(-1) = 4(-1)^3 + 7(-1)^2 - 2(-1) - 5$
$p(-1) = 4(-1) + 7(1) + 2 - 5$
$p(-1) = -4 + 7 + 2 - 5$
$p(-1) = 0$
Since $p(-1) = 0$,$(x+1)$ is indeed a factor of the given polynomial.

(A) To factorise the expression $x^{3}-125$,we use the algebraic identity for the difference of two cubes: $a^{3}-b^{3} = (a-b)(a^{2}+ab+b^{2})$.
Here,$x^{3}-125$ can be written as $x^{3}-5^{3}$.
Comparing this with $a^{3}-b^{3}$,we have $a = x$ and $b = 5$.
Substituting these values into the identity: $x^{3}-5^{3} = (x-5)(x^{2} + x(5) + 5^{2})$.
Simplifying the expression,we get: $(x-5)(x^{2}+5x+25)$.