Practice Questions

Identify the degree of the polynomial $p(x) = 7x^3 - 4x^5 + 2x + 9$.

Apply a suitable identity to find the product of $(3x - 5)(3x + 4)$.

Name the type of polynomial $p(t) = 5t - \sqrt{7}$ based on its degree.

Apply the Factor Theorem to determine if $g(x) = x - 3$ is a factor of the polynomial $p(x) = x^3 - 4x^2 + x + 6$.

Solve for the factors of the expression $25x^2 - 20xy + 4y^2$ using an appropriate identity.

Define a polynomial in one variable $x$.

Identify the terms in the polynomial $3y^2 + 5y - 7$.

Recall the algebraic identity for $(x-y)^2$.

A student claims that $f(t) = 3t^2 - \sqrt{5}t + t^{-1}$ is a polynomial in one variable. Critique this statement and justify your conclusion.

Formulate an argument to critique the statement: 'If a polynomial has three terms, it must be a cubic polynomial.' Provide a counterexample.

Propose a polynomial in one variable that is a binomial, quadratic, and has a coefficient of $\sqrt{3}$ for its linear term.

Compare the polynomials $p(x) = 7x^3 - 5x^4 + 3x + 9$ and $q(x) = 4x^3 - 8x + 2$ based on their degree, number of terms, and the coefficient of the $x^3$ term.

Solve for the value of $k$ if $x-2$ is a factor of the polynomial $p(x) = 2x^2 + kx - 6$.

Calculate the value of the polynomial $p(y) = 4y^3 - 3y^2 + 5y - 6$ at $y = -\frac{1}{2}$.

Solve for the zeroes of the polynomial $p(x) = x^2 + 2x - 8$ and demonstrate that these values satisfy the polynomial equation $p(x)=0$.

Evaluate the expression $997^3$ by designing a calculation that uses the identity $(x-y)^3 = x^3 - 3x^2y + 3xy^2 - y^3$. Justify your choice of $x$ and $y$.

Propose a method using the Factor Theorem to justify that $(x-a)$ is a factor of $p(x) = x^n - a^n$ for any positive integer $n$.

If $x-1$ is a factor of the polynomial $p(x) = k^2x^2 - 3kx + 3k - 1$, formulate the possible values of $k$.

Explain which of the following expressions are polynomials in one variable. State the reason if an expression is not a polynomial. (i) $3t^2 - \sqrt{5}t + 1$ (ii) $y + \frac{3}{y}$ (iii) $x^{10} + y^5 + 2$

Use a suitable identity to find the product of $(x+8)(x-10)$.

Find the value of the polynomial $p(x) = 5x^2 - 3x + 7$ at $x = -1$.

Explain the difference between a monomial, a binomial, and a trinomial. Provide one example for each.

List the coefficients of $x^3$, $x^2$, and $x$ in the polynomial $p(x) = 2 - x^2 + 5x^3$.

Define what a 'zero of a polynomial' is. Then, find the zero of the linear polynomial $p(x) = 3x - 2$.

Examine if $x=1$ and $x=-2$ are zeroes of the cubic polynomial $p(x) = x^3 + 2x^2 - 5x - 6$.

Describe how to classify polynomials based on their degree. List the names for polynomials of degree one, two, and three, and provide the general form for each in one variable $x$.

Calculate the value of $(103)^3$ by applying a suitable algebraic identity.

Demonstrate the expansion of $(\frac{1}{2}x - y + 3z)^2$ using a suitable identity.

What is the degree of a non-zero constant polynomial?

Create a polynomial expression for the volume of a cuboid whose dimensions are $(x-2)$, $(x+1)$, and $(x+3)$. Then, evaluate the volume for $x=2$ and justify the result in the context of the cuboid's dimensions.

Formulate a quadratic polynomial with rational coefficients, given that one of its zeroes is $\frac{1}{2+\sqrt{3}}$.

Analyze if the polynomial $g(x) = 2x - 1$ is a factor of $p(x) = 4x^3 + 4x^2 - x - 1$. Explain the implication of your result.

Create a cubic polynomial in variable $x$ whose zeroes are $2, -1,$ and $3$. Justify your formulation process.

Evaluate the two methods—splitting the middle term and using the Factor Theorem—to factorise the polynomial $p(x) = 2x^2 + 7x + 3$. Justify which method is more efficient for this polynomial.

Critique the proposed factorization of $4x^2+9y^2+16z^2+12xy-24yz-16xz$ as $(2x+3y+4z)^2$. Identify the error in the proposal and formulate the correct factorization.

Analyze the following expressions to determine which are polynomials in one variable. Provide a reason for each expression that is not a polynomial. (i) $3y^2 - 5y + \sqrt{7}$ (ii) $z + \frac{3}{z}$ (iii) $x^{12} + y^3 + t^{20}$ (iv) $5\sqrt{t} + t^2$

Without performing direct calculation of the cubes, solve for the value of $(-10)^3 + (7)^3 + (3)^3$.

Recall the identity for $(x+y+z)^2$ and use it to expand $(2a-b+c)^2$.

Solve for the factors of the expression $8x^3 - y^3 - 12x^2y + 6xy^2$ by applying a suitable identity.

Explain the Factor Theorem. Use it to determine if $g(x) = x-3$ is a factor of the polynomial $p(x) = x^3 - 4x^2 + x + 6$.

Propose a method to factorise the expression $a^6 - b^6$ in two different ways using the identities for difference of squares and difference of cubes. Justify that both methods lead to the same complete factorisation.

Solve for the factors of the cubic polynomial $p(x) = x^3 + 6x^2 + 11x + 6$ by using the Factor Theorem.

Formulate a cubic polynomial $p(x)$ such that its constant term is $10$, the sum of its coefficients is $20$, and $p(-1)=0$.

Design a quadratic polynomial $p(x) = ax^2 + bx + c$ where the coefficients $a, b, c$ are integers, $a \neq 0$, one of its zeroes is $3 + \sqrt{2}$, and $p(1) = -4$.

Justify the statement: 'The Factor Theorem is a special case of the Remainder Theorem.'