Polynomials in ℂ practice questions

Question 1

The degree of the polynomial $P(X) = 2X^4 - 5X^3 + X^2 - 7$ is:

Accepted Answer

The degree is the largest exponent of $X$ with a non-zero coefficient. Here it is $4$.

Answer

$2$

Answer

$3$

Answer

$4$

Answer

$7$

Question 2

For $P(X) = X^3 - 2X + 5$, the value of $P(2)$ is:

Accepted Answer

$P(2) = 8 - 4 + 5 = 9$.

Answer

$5$

Answer

$7$

Answer

$9$

Answer

$13$

Question 3

For $A = X^2 + 3X - 1$ and $B = 2X^2 - X + 4$, the polynomial $A + B$ equals:

Accepted Answer

Add coefficients: $(1+2)X^2 + (3-1)X + (-1+4) = 3X^2 + 2X + 3$.

Answer

$X^2 + 2X + 3$

Answer

$3X^2 + 4X + 5$

Answer

$3X^2 + 2X + 3$

Answer

$-X^2 + 4X - 5$

Question 4

The remainder of the division of $P(X) = X^3 + 2X^2 - X + 5$ by $X - 1$ is:

Accepted Answer

$P(1) = 1 + 2 - 1 + 5 = 7$.

Answer

$0$

Answer

$5$

Answer

$7$

Answer

$9$

Question 5

The polynomial $P(X) = X^3 - X^2 - 4X + 4$ factors as:

Accepted Answer

$P(1) = 0$, so $X - 1$ divides $P$. By Horner: $P(X) = (X - 1)(X^2 - 4) = (X - 1)(X - 2)(X + 2)$.

Answer

$(X - 1)(X - 2)(X + 2)$

Answer

$(X + 1)(X - 2)(X + 2)$

Answer

$(X - 1)^2(X + 4)$

Answer

$(X - 1)(X + 2)^2$

Question 6

The quotient when $P(X) = X^3 - 4X^2 + 5X - 2$ is divided by $X - 2$ has degree:

Accepted Answer

$\deg(Q) = 3 - 1 = 2$.

Answer

$1$

Answer

$2$

Answer

$3$

Answer

$4$

Question 7

For the polynomial $P(X) = 2X^3 - 6X^2 + X - 4$ with roots $x_1, x_2, x_3$, the product $x_1 x_2 x_3$ equals:

Accepted Answer

$x_1 x_2 x_3 = -\dfrac{-4}{2} = 2$.

Answer

$-4$

Answer

$-2$

Answer

$2$

Answer

$4$

Question 8

For $P(X) = X^3 - 6X^2 + 11X - 6$ with roots $x_1, x_2, x_3$, the value of $x_1 + x_2 + x_3$ is:

Accepted Answer

The leading coefficient is $1$ and the coefficient of $X^2$ is $-6$, so $x_1 + x_2 + x_3 = -(-6) = 6$. (The roots are in fact $1, 2, 3$.)

Answer

$-11$

Answer

$-6$

Answer

$6$

Answer

$11$

Question 9

Dividing $x^{2} - 5x + 6$ by $(x - 2)$ gives the quotient:

Accepted Answer

$x^{2} - 5x + 6 = (x - 2)(x - 3)$.

So $\dfrac{x^{2} - 5x + 6}{x - 2} = x - 3$, with no remainder.

Answer

$x - 6$

Answer

$x + 3$

Answer

$x - 3$

Answer

$x - 2$

Question 10

Factor over $\mathbb{R}$: $x^{2} - 9$.

Accepted Answer

$x^{2} - 9 = x^{2} - 3^{2} = (x - 3)(x + 3)$ — the difference-of-squares identity.

Answer

$(x - 3)(x + 3)$

Answer

$(x - 3)^{2}$

Answer

$(x + 3)^{2}$

Answer

$(x - 9)(x + 1)$

Polynomials in ℂ

Problems & worked solutions

Problem 1 — Polynomials in ℂ

Problem 2 — Polynomials in ℂ

Problem 3 — Polynomials in ℂ

Problem 4 — Polynomials in ℂ

Problem 5 — Polynomials in ℂ

Problem 6 — Polynomials in ℂ

Problem 7 — Polynomials in ℂ

Problem 8 — Polynomials in ℂ

Problem 9 — Division, Bezout, Horner

Problem 10 — Factorization in C and R

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