Grade 10 Maths · Algebra

Collect like terms

Add and subtract terms with the same letters and powers to simplify an expression (for example 3x + 2y − x + 4y = 2x + 6y).

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Algebra Grade 9: Add and Subtract like termsKevinmathscience

ALGEBRA - Add and Subtract LIKE TERMS - Grade 8SHFT - Your Learning

Combining like terms introductionKhan Academy

Combining Like Terms Using The Distributive Property - AlgebraThe Organic Chemistry Tutor

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Question 1.3 · May/June 2025 · Paper 1 · National (DBE) · 3 marks

1.3 Show that

(1 + x^{m} + x^{- n})^{2} - (1 - x^{m} - x^{- n})^{2}

is divisible by

2

for all real values of

m

and

n

.(3)

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

More past paper questions: collect like terms

Work each one out first, then open the memo and check every tick.

3. Question 6.2.1 · May/June 2025 · Paper 2 · National (DBE) · 2 marks
6.2 Consider the expression: $sin (A - B) - sin (A + B)$
6.2.1 Prove that $sin (A - B) - sin (A + B) = - 2 cos A sin B$ (2)
Show the memo (2 ticks)Memo
1. ✓
  Expand both terms using compound angle identities $L H S = sin (A - B) - sin (A + B)$
  $= (sin A cos B - cos A sin B) - (sin A cos B + cos A sin B)$
  1 mark · Use a compound angle identity
2. ✓
  Distribute negative sign and simplify $= sin A cos B - cos A sin B - sin A cos B - cos A sin B$
  $= - 2 cos A sin B$
  $= R H S$
  1 mark · Collect like terms
4. Question 9.5 · May/June 2025 · Paper 1 · National (DBE) · 6 marks
Given: $f (x) = x^{3} - 6 x^{2} + 9 x - 4 = (x - 4) (x - k)^{2}$
9.5 Calculate the maximum vertical distance between $f$ and $h$ for $1 < x < 3$ , if $h (x) = - 2 f^{'} (x)$ .(6)
Show the memo (6 ticks)Memo
1. ✓
  Establish the expression for the distance $h (x) = - 2 (3 x^{2} - 12 x + 9) = - 6 x^{2} + 24 x - 18$
  $Distance = h (x) - f (x)$
  $= (- 6 x^{2} + 24 x - 18) - (x^{3} - 6 x^{2} + 9 x - 4)$
  1 mark · Find a horizontal or vertical length
2. ✓
  Simplify the distance expression $= - 6 x^{2} + 24 x - 18 - x^{3} + 6 x^{2} - 9 x + 4$
  $= - x^{3} + 15 x - 14$
  1 mark · Collect like terms
3. ✓
  Differentiate the distance expression and equate to 0To find max distance, let $D^{'} (x) = 0$ .
  $D^{'} (x) = - 3 x^{2} + 15 = 0$
  1 mark · Find stationary points
4. ✓
  Solve the derivative equation for x squared $3 x^{2} = 15$
  $x^{2} = 5$
  1 mark · Solve a linear equation
5. ✓
  Find the correct x-value within the given domain $x = \pm 5$
  But the domain is restricted to $1 < x < 3$ .
  $∴ x = 5$ (since $- 5$ is not in the interval)
  1 mark · Solve a quadratic by factorising
6. ✓
  Calculate the maximum vertical distanceSubstitute $x = 5$ into the distance equation:
  $Max distance = - (5)^{3} + 15 (5) - 14$
  $= - 55 + 155 - 14$
  $= 105 - 14 \approx 8, 36$
  1 mark · Find stationary points
5. Question 1.3.3 · March 2025 · Provincial · 4 marks
1.3 Simplify the following fully:
1.3.3 $\frac{3}{x ^{2} - 9} + \frac{2}{x - 3} - \frac{1}{x + 3}$ (4)
Show the memo (4 ticks)Memo
1. ✓
  Factorise denominator $\frac{3}{( x - 3 ) ( x + 3 )} + \frac{2}{x - 3} - \frac{1}{x + 3}$
  1 mark · Factorise a difference of squares
2. ✓
  Find LCD and combine $\frac{3 + 2 ( x + 3 ) - 1 ( x - 3 )}{( x - 3 ) ( x + 3 )}$
  1 mark · Add or subtract algebraic fractions
3. ✓
  Simplify numerator $3 + 2 x + 6 - x + 3 = x + 12$
  1 mark · Collect like terms
4. ✓
  Final Answer $\frac{x + 12}{( x - 3 ) ( x + 3 )}$
  1 mark · Collect like terms
6. Question 1.2.1 · March 2020 · Provincial · 3 marks
1.2 Simplify the following expressions fully:
1.2.1 $(2 x - 1) (x^{2} - 3 x + 1)$ (3)
Show the memo (2 ticks)Memo
1. ✓
  Expansion $2 x^{3} - 6 x^{2} + 2 x - x^{2} + 3 x - 1$
  1 mark · Multiply out larger brackets
2. ✓
  Simplify $2 x^{3} - 7 x^{2} + 5 x - 1$
  2 marks · Collect like terms

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