CAT Polynomials, higher-degree equations & binomial expansion Questions

Polynomial problems connect factors, roots and coefficients. Higher-degree equations can often be reduced through factorisation or substitution, while binomial expansion tracks the terms of a power.

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Key concepts

• Factor and remainder theorems
• Roots and coefficients
• Binomial coefficients and general terms

CAT polynomials, higher-degree equations & binomial expansion practice questions

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Common mistakes

• Losing roots when cancelling a variable factor.
• Using the term number as the exponent without checking the expansion's indexing.

Practical solving approaches

• Test simple roots and look for repeated algebraic structures before expanding.
• Write the general term of a binomial expansion and match the required power.

Polynomials, higher-degree equations & binomial expansion FAQs

Should I expand every polynomial?

Not necessarily. Factorisation, symmetry or a substitution can preserve useful structure and avoid lengthy calculations.

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CAT Polynomials, higher-degree equations & binomial expansion Questions

Polynomial problems connect factors, roots and coefficients. Higher-degree equations can often be reduced through factorisation or substitution, while binomial expansion tracks the terms of a power.

5 free practice questions

Key concepts

• Factor and remainder theorems
• Roots and coefficients
• Binomial coefficients and general terms

CAT polynomials, higher-degree equations & binomial expansion practice questions

Question 1Multiple choiceMEDIUM

Question 1

Select the correct alternative from the given choices

Answer the questions independently of each other.

Find the remainder when x³ – 5x² + 7x – 9 is divided by x² + x – 6.

Choose an answer

A.19x – 45B.19x – 38C.19x – 49D.19x – 31

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Question 2Multiple choiceEASY

Question 2

Select the correct alternative from the given choices.

Answer the questions independently of each other.

The roots of the equation x(x – 2)(3x – 7) = 4 are necessarily,

Choose an answer

A.Positive integersB.Real, positive numbersC.Real, negative numbersD.Whole numbers

Check answerView solution

Question 3Multiple choiceHARD

Question 3

Select the correct alternative from the given choices.

Answer the questions independently of each other.

If f(x) = x³ – 4x + p, and f(0) and f(1) are of opposite signs, then which of the following is necessarily true.

Choose an answer

A.–1 < p < 2B.0 < p < 3C.–2 < p < 1D.–3 < p < 0

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Question 4Type in the answerMEDIUM

Question 4

Type in your answer in the input box provided below the question.

Answer the questions independently of each other.

If a + 2b = 10 and ab = 15, find the value of a³ + 8b³.

Your answer

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Question 5Multiple choiceMEDIUM

Question 5

Select the correct alternative from the given choices.

Answer the questions independently of each other.

For which of the following values y⁴- 10y³ + 5y² + 100y + 100 = 0?

Choose an answer

A.

B.

C.

D.

Check answerView solution

Common mistakes

• Losing roots when cancelling a variable factor.
• Using the term number as the exponent without checking the expansion's indexing.

Practical solving approaches

• Test simple roots and look for repeated algebraic structures before expanding.
• Write the general term of a binomial expansion and match the required power.

Polynomials, higher-degree equations & binomial expansion FAQs

Should I expand every polynomial?

Not necessarily. Factorisation, symmetry or a substitution can preserve useful structure and avoid lengthy calculations.

More Algebra practice

• Functions & graphs →
• Inequalities & modulus →
• Linear equations & word problems →
• Logarithms, indices & surds →
• Quadratic equations & expressions →
• Sequences & series →