Grade 12 Maths · Calculus · Use the remainder theorem

Use the remainder theorem Grade 12

Find the remainder when a polynomial is divided by (x - a) by working out f(a), or use a given remainder to find an unknown coefficient, for example f(x) = x^3 + kx - 4 divided by x - 2 leaves 6, so 8 + 2k - 4 = 6 and k = 1.

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What is the Remainder theorem? | Grade 12 MathMatheNatic

Remainder & Factor Theorems | Grade 12 Calculus Part 7Maths With Mudinda

Remainder Theorem - Polynomial with Unknown Coefficient - Exam QuestionRadford Mathematics

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Question 8.2 · May/June 2023 · Paper 1 · National (DBE) · 2 marks

Given:

f (x) = x^{3} + 4 x^{2} - 7 x - 10

8.2 Show that

2

is a root of the equation

f (x) = 0

.(2)

Work it out on paper first. Then mark yourself with the official memo, one tick at a time.

More questions: use the remainder theorem

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3. Practice · Siyavula Grade 12 Maths · Ex 5.4 · 3 marks
Calculate the value of $m$ if $2 x^{3} - 7 x^{2} + m x - 26$ is divided by $x - 2$ and gives a remainder of $- 24$ .(3)
Show the memo (3 ticks)Memo
1. ✓
  Substitute x = 2 $a (2) = 2 (2)^{3} - 7 (2)^{2} + m (2) - 26$
  $= 16 - 28 + 2 m - 26 = 2 m - 38$
  1 mark · Use the remainder theorem
2. ✓
  Equate to the remainder $2 m - 38 = - 24$
  1 mark · Use the remainder theorem
3. ✓
  Value of m $2 m = 14$
  $m = 7$
  1 mark · Use the remainder theorem
Question adapted from Siyavula Education (CC BY 3.0) · Solution by NavyBlue
4. Practice · Siyavula Grade 12 Maths · Ex 5.4 · 3 marks
Determine the value of $t$ if $x^{3} + t x^{2} + 8 x + 21$ divided by $x + 1$ gives a remainder of $16$ .(3)
Show the memo (3 ticks)Memo
1. ✓
  Substitute x = -1Let $a (x) = x^{3} + t x^{2} + 8 x + 21$
  $a (- 1) = (- 1)^{3} + t (- 1)^{2} + 8 (- 1) + 21 = t + 12$
  1 mark · Use the remainder theorem
2. ✓
  Equate to the remainder $t + 12 = 16$
  1 mark · Use the remainder theorem
3. ✓
  Value of t $t = 4$
  1 mark · Use the remainder theorem
Question adapted from Siyavula Education (CC BY 3.0) · Solution by NavyBlue
5. Practice · Siyavula Grade 12 Maths · Ex 5.7 · 2 marks
(a) Show that $x - 2$ is a factor of $3 x^{3} - 11 x^{2} + 12 x - 4$ .(2)
Show the memo (2 ticks)Memo
1. ✓
  Evaluate at x = 2 $a (2) = 3 (8) - 11 (4) + 12 (2) - 4$
  $= 24 - 44 + 24 - 4 = 0$
  1 mark · Use the remainder theorem
2. ✓
  Conclusion $a (2) = 0$ , so $x - 2$ is a factor
  1 mark · Use the remainder theorem
Question adapted from Siyavula Education (CC BY 3.0) · Solution by NavyBlue
6. Practice · Siyavula Grade 12 Maths · Ex 5.4 · 2 marks
Use the remainder theorem to determine the remainder $R$ when $g (x)$ is divided by $h (x)$ :
(a) \begin{align*}
g(x) &= x^{3} + 4x^{2} + 11x - 5 \\
h(x) &= x - 1
\end{align*}(2)
Show the memo (2 ticks)Memo
1. ✓
  Value to substitute $x - 1 = 0 \Rightarrow x = 1$
  $R = g (1) = (1)^{3} + 4 (1)^{2} + 11 (1) - 5$
  1 mark · Use the remainder theorem
2. ✓
  Remainder $= 1 + 4 + 11 - 5$
  $R = 11$
  1 mark · Use the remainder theorem
Question adapted from Siyavula Education (CC BY 3.0) · Solution by NavyBlue

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