Open Practice Tests

Equivalent expressions and polynomials — Topic practice

5 exercises · 2 automatically checked · 3 written self-checks · Untimed

Instructions and review guidance

2 questions · Shuffled order · Untimed practice

Work through the set before checking solutions. Guide question labels match the source lesson references. 3 written responses appear below the question bank, with model answers for self-review; these are not automatically graded.

Guide question labels stay the same when the order changes. Check answers when you are ready to review.

Written self-checks are below the question bank and are excluded from the automatic score.

Written responses are self-review exercises.

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The drill

Question 3

  • Guide question E1.3

For all real x, (x + a)(x + 4) = x2 + 9x + b. Find a + b.

Enter an integer, decimal or fraction without units. Use up to 5 characters, or 6 including a leading minus.

Question 5

  • Guide question E1.5

If u + v = 9 and uv = 14, find u2 + v2.

Enter an integer, decimal or fraction without units. Use up to 5 characters, or 6 including a leading minus.

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Written self-checks

Write your complete response before opening the model answer. Compare both your answer and your reasoning.

Guide question E1.1

Simplify (2x2 − 3x + 1) − (5x2 + x − 6).

Show model answerHide model answer for question E1.1

−3x2 − 4x + 7

Distribute the subtraction to every term in the second polynomial:

(2x2 − 3x + 1) − (5x2 + x − 6) = 2x2 − 3x + 1 − 5x2 − x + 6.

Then combine like terms. The constant becomes 1 + 6 = 7, not 1 − 6. Review 1.01.

Guide question E1.2

Factor 8x2 − 2 completely over the reals.

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2(2x − 1)(2x + 1)

Factor out the greatest common factor before looking for a pattern:

8x2 − 2 = 2(4x2 − 1) = 2((2x)2 − 12) = 2(2x − 1)(2x + 1).

A difference of squares produces conjugate factors, not a repeated factor. Expanding checks the result. Review 1.03 and 1.04.

Guide question E1.4

Simplify x2 − 16x + 4 and state the original domain restriction.

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x − 4, with x ≠ −4

The original denominator requires x ≠ −4. On that domain,

x2 − 16x + 4 = (x − 4)(x + 4)x + 4 = x − 4.

The canceled expression is still undefined at −4. A simplified rule does not restore a missing input. Review 1.08.