Arithmetic and terms

Terms and brackets

Simplifying terms is the basic skill behind every equation. München's T-Kurs test names algebraic fractions (Bruchterme) explicitly, and they all start with clean expanding and collecting.

München Named for the T-Kurs maths test at Studienkolleg München. Source

The rules

• Expand: multiply every term in one bracket by every term in the other: (a + b)(c + d) = ac + ad + bc + bd.
• Collect like terms: only terms with the same variable and power can be added.
• A minus sign before a bracket changes every sign inside it.
• Algebraic fractions: factorise numerator and denominator first, then cancel common factors.

Practice questions with solutions

Generated by the same engine as the Kollego course and solvable without a calculator. Pick an answer in your head, then open the solution.

1. 1. Expand (x − 5)(3x + 3). What is the constant term?

   • a-14
   • b-15
   • c-12
   • d3
   Show solution

   Answer: -15

   (ax + p)(cx + q) = ac·x² + (aq + pc)x + pq.

   Constant term = p·q = -5·3 = -15.

2. 2. Simplify 7x + 3 + 4x + 5, then evaluate at x = -2.

   • a-14
   • b-15
   • c-13
   • d-22
   Show solution

   Answer: -14

   Combine like terms first, then substitute the value of x.

   Simplified: 11x + 8. At x = -2: 11·(−2) + 8 = -22 + 8 = -14.

3. 3. Expand (2x − 3)(2x − 5). What is the coefficient of x?

   • a-15
   • b15
   • c-16
   • d4
   Show solution

   Answer: -16

   (ax + p)(cx + q) = ac·x² + (aq + pc)x + pq.

   Coefficient of x = a·q + p·c = 2·(−5) − 3·2 = -16.

4. 4. Expand (3x + 4)(3x + 3). What is the coefficient of x?

   • a9
   • b21
   • c12
   • d22
   Show solution

   Answer: 21

   (ax + p)(cx + q) = ac·x² + (aq + pc)x + pq.

   Coefficient of x = a·q + p·c = 3·3 + 4·3 = 21.

5. 5. Expand (x − 6)(2x + 3). What is the constant term?

   • a2
   • b-17
   • c-9
   • d-18
   Show solution

   Answer: -18

   (ax + p)(cx + q) = ac·x² + (aq + pc)x + pq.

   Constant term = p·q = -6·3 = -18.

6. 6. Simplify by combining like terms: 6x + 1 + 3x − 5 What is the coefficient of x in the simplified expression?

   • a3
   • b8
   • c9
   • d10
   Show solution

   Answer: 9

   Combine terms with the same variable part: add the x-coefficients, and separately add the constants.

   x-terms: 6x + 3x = 9x. constants: 1 − 5 = -4. Simplified: 9x − 4.

Common mistakes

• Forgetting to change all signs after a minus before a bracket.
• Cancelling terms instead of factors: (x + 2)/2 is not x + 1.
• Missing one product when expanding two brackets.

Related topics

All maths topics · T-Kurs · M-Kurs · W-Kurs