Polynomial Basics — Practice Problems and Worked Solutions

The formulas you need here

16 worked problems

Every step is generated and checked by a computer algebra system.

1. 1.

   Identify the degree of the polynomial.

   $$3x^2-2x+7$$

   Answer: $2$

   Step-by-step solution

   1. $$\text{Find the greatest exponent attached to }x.$$
   2. $$\mathrm{deg}(3x^2-2x+7)=2$$
2. 2.

   Identify the leading coefficient.

   $$-2x^3+5x-1$$

   Answer: $-2$

   Step-by-step solution

   1. $$\text{Write terms from greatest exponent to least.}$$
   2. $$\text{The leading term is }-2x^3\text{, so its coefficient is }-2.$$
3. 3.

   Simplify the polynomial expression.

   $$3x-2$$

   Answer: $3x-2$

   Step-by-step solution

   1. $$\text{Organize terms by their exponent.}$$
   2. $$\text{Distribute if needed, then combine like powers.}$$
   3. $$=3x-2$$
4. 4.

   Simplify the polynomial expression.

   $$3x^2+3x$$

   Answer: $3x\left(x+1\right)$

   Step-by-step solution

   1. $$\text{Organize terms by their exponent.}$$
   2. $$\text{Distribute if needed, then combine like powers.}$$
   3. $$=3x^2+3x$$
5. 5.

   Simplify the polynomial expression.

   $$x\left(x+4\right)$$

   Answer: $x\left(x+4\right)$

   Step-by-step solution

   1. $$\text{Organize terms by their exponent.}$$
   2. $$\text{Distribute if needed, then combine like powers.}$$
   3. $$=x^2+4x$$
6. 6.

   Simplify the polynomial expression.

   $$\left(x+2\right)\left(x+3\right)$$

   Answer: $x^2+5x+6$

   Step-by-step solution

   1. $$\text{Organize terms by their exponent.}$$
   2. $$\text{Distribute if needed, then combine like powers.}$$
   3. $$=x^2+5x+6$$
7. 7.

   Simplify the polynomial expression.

   $$5x-3$$

   Answer: $5x-3$

   Step-by-step solution

   1. $$\text{Organize terms by their exponent.}$$
   2. $$\text{Distribute if needed, then combine like powers.}$$
   3. $$=5x-3$$
8. 8.

   Simplify the polynomial expression.

   $$3x^2+3x$$

   Answer: $3x\left(x+1\right)$

   Step-by-step solution

   1. $$\text{Organize terms by their exponent.}$$
   2. $$\text{Distribute if needed, then combine like powers.}$$
   3. $$=3x^2+3x$$
9. 9.

   Simplify the polynomial expression.

   $$x\left(x+5\right)$$

   Answer: $x\left(x+5\right)$

   Step-by-step solution

   1. $$\text{Organize terms by their exponent.}$$
   2. $$\text{Distribute if needed, then combine like powers.}$$
   3. $$=x^2+5x$$
10. 10.

    Simplify the polynomial expression.

    $$\left(x+1\right)\left(x+4\right)$$

    Answer: $x^2+5x+4$

    Step-by-step solution

    1. $$\text{Organize terms by their exponent.}$$
    2. $$\text{Distribute if needed, then combine like powers.}$$
    3. $$=x^2+5x+4$$
11. 11.

    Simplify the polynomial expression.

    $$5x+5$$

    Answer: $5x+5$

    Step-by-step solution

    1. $$\text{Organize terms by their exponent.}$$
    2. $$\text{Distribute if needed, then combine like powers.}$$
    3. $$=5x+5$$
12. 12.

    Simplify the polynomial expression.

    $$2x\left(x-3\right)$$

    Answer: $2x\left(x-3\right)$

    Step-by-step solution

    1. $$\text{Organize terms by their exponent.}$$
    2. $$\text{Distribute if needed, then combine like powers.}$$
    3. $$=2x^2-6x$$
13. 13.

    Simplify the polynomial expression.

    $$\left(x-2\right)\left(x+5\right)$$

    Answer: $x^2+3x-10$

    Step-by-step solution

    1. $$\text{Organize terms by their exponent.}$$
    2. $$\text{Distribute if needed, then combine like powers.}$$
    3. $$=x^2+3x-10$$
14. 14.

    Simplify the polynomial expression.

    $$4x^2-4x$$

    Answer: $4x\left(x-1\right)$

    Step-by-step solution

    1. $$\text{Organize terms by their exponent.}$$
    2. $$\text{Distribute if needed, then combine like powers.}$$
    3. $$=4x^2-4x$$
15. 15.

    Simplify the polynomial expression.

    $$\left(x+1\right)\left(2x+3\right)$$

    Answer: $2x^2+5x+3$

    Step-by-step solution

    1. $$\text{Organize terms by their exponent.}$$
    2. $$\text{Distribute if needed, then combine like powers.}$$
    3. $$=2x^2+5x+3$$
16. 16.

    Simplify the polynomial expression.

    $$3x\left(x+2\right)$$

    Answer: $3x\left(x+2\right)$

    Step-by-step solution

    1. $$\text{Organize terms by their exponent.}$$
    2. $$\text{Distribute if needed, then combine like powers.}$$
    3. $$=3x^2+6x$$

More from Math Foundations

• Order of Operations
• Signed Numbers
• Fraction Sense
• Ratios and Percentages
• Exponents and Roots
• Algebraic Expressions
• Linear Equations
• Inequalities
• Coordinate Plane
• Reading Graphs
• Function Machines
• Substitution
• Factoring Basics