Exam Response (polynomials)

Exam Response (polynomials)

What is it?

Typical tasks: remainder by substituting; prove a factor; factor a cubic; synthetic table; Vieta sum; find k. Write p(c) as its own line. If you claim (x-c) is a factor, show remainder 0.

In simple words

• Typical tasks: remainder by substituting; prove a factor; factor a cubic; synthetic table; Vieta sum; find k.
• Write p(c) as its own line.
• If you claim (x-c) is a factor, show remainder 0.
• x+3 is x-(-3). Plug -3, not 3.

Important terms

EXAM RESPONSE (POLYNOMIALS): Typical tasks: remainder by substituting; prove a factor; factor a cubic; synthetic table; Vieta sum; find k. Write p(c) as its own line. If you claim (x-c) is a factor, show remainder 0.

1. REMAINDER / FACTOR: Identify c from x-c. Compute p(c). Zero means factor. Non-zero is the remainder.

2. FACTORISE: Find one root. Synthetic or long divide. Quadratic formula. Multiplicity if a repeated factor. Vieta to check.

What you should learn

• State the main ideas and notation used in exam response (polynomials).
• Apply the rules of exam response (polynomials) in a clear, step-by-step solution.
• Check signs, conditions, and the final answer for mathematical accuracy.

Examples

• x+2 -> p(-2)
• Show the arithmetic
• State remainder

REMAINDER / FACTOR

Identify c from x-c. Compute p(c). Zero means factor. Non-zero is the remainder.

FACTORISE

Find one root. Synthetic or long divide. Quadratic formula. Multiplicity if a repeated factor. Vieta to check.

Full school-level explanation

Typical tasks: remainder by substituting; prove a factor; factor a cubic; synthetic table; Vieta sum; find k. Write p(c) as its own line. If you claim (x-c) is a factor, show remainder 0.

Substitute, then divide:

WHAT — EXAM RESPONSE (POLYNOMIALS)

Typical tasks: remainder by substituting; prove a factor; factor a cubic; synthetic table; Vieta sum; find k. Write p(c) as its own line. If you claim (x-c) is a factor, show remainder 0.

RULE — 1. REMAINDER / FACTOR

Identify c from x-c. Compute p(c). Zero means factor. Non-zero is the remainder.

METHOD — 2. FACTORISE

Find one root. Synthetic or long divide. Quadratic formula. Multiplicity if a repeated factor. Vieta to check.

WORKED EXAMPLE

Worked 4-mark remainder. divisor x+3 means c=-3 p(x)=2x^3 - x + 4 p(-3)=2(-27) -(-3) +4 = -54+3+4=-47 remainder -47

REMAINDER / FACTOR

Identify c from x-c. Compute p(c). Zero means factor. Non-zero is the remainder.

Remember

• c from the divisor

Examples

• x+2 -> p(-2)
• Show the arithmetic
• State remainder

FACTORISE

Find one root. Synthetic or long divide. Quadratic formula. Multiplicity if a repeated factor. Vieta to check.

Remember

• one root then divide

Examples

• Guess 1,-1,2
• Quotient quadratic
• Expand check

How to answer a exam response (polynomials) question

How it works

• Identify the given quantities, notation, and required result.
• Choose the matching rule and write it before substitution.
• Work one algebraic step at a time, keeping brackets and signs clear.
• Check restrictions, simplify the result, and verify it where possible.

Remember

• remainder = p(c)
• factor iff p(c)=0
• -47
• sign of c
• c from the divisor
• one root then divide
• remainder = p(c)
• factor iff p(c)=0

How the ideas connect

Worked 4-mark remainder. divisor x+3 means c=-3 p(x)=2x^3 - x + 4 p(-3)=2(-27) -(-3) +4 = -54+3+4=-47 remainder -47

Deeper understanding

Go beyond memorising Exam Response (polynomials). Connect the definition to reasons, relationships, exceptions, and exam wording.

How the ideas connect

Worked 4-mark remainder. divisor x+3 means c=-3 p(x)=2x^3 - x + 4 p(-3)=2(-27) -(-3) +4 = -54+3+4=-47 remainder -47

Common mistakes

Exam focus

Core Idea

1. REMAINDER / FACTOR

2. FACTORISE

Worked Example

Exam focus

Quick revision

• remainder = p(c)
• factor iff p(c)=0
• -47
• sign of c
• EXAM RESPONSE (POLYNOMIALS): Typical tasks: remainder by substituting; prove a factor; factor a cubic; synthetic table; Vieta sum
• 1. REMAINDER / FACTOR: Identify c from x-c. Compute p(c). Zero means factor. Non-zero is the remainder.

x+3 is x-(-3). Plug -3, not 3.

For long answers on Exam Response (polynomials): define clearly → explain the main parts → give one example → state one exam caution.