9 Best Multiplying Polynomials Worksheet For Students - The Teach ... - Free Printable

Here are the step-by-step solutions for each problem on the worksheet.

1. Multiply $4x$ with $3x$.
* Multiply the numbers (coefficients): $4 \times 3 = 12$.
* Multiply the variables: $x \times x = x^2$.
* Combine them: $12x^2$

2. What will be the result of the product of $(px^2)$ and $(px^2 + x + p)$?
* Distribute $(px^2)$ to every term inside the parentheses.
* First term: $(px^2) \cdot (px^2) = p \cdot p \cdot x^2 \cdot x^2 = p^2x^4$
* Second term: $(px^2) \cdot (x) = px^3$
* Third term: $(px^2) \cdot (p) = p^2x^2$
* Combine them: $p^2x^4 + px^3 + p^2x^2$

3. Find the product of the following polynomials:

A. $(-4x^2 - 2)$ and $(3x + 4)$
* Use the FOIL method (First, Outer, Inner, Last).
* First: $(-4x^2)(3x) = -12x^3$
* Outer: $(-4x^2)(4) = -16x^2$
* Inner: $(-2)(3x) = -6x$
* Last: $(-2)(4) = -8$
* Combine them: $-12x^3 - 16x^2 - 6x - 8$

B. $(-z^3 - 2z^2)$ and $(5z + z^2)$
* Distribute each term from the first group to the second group.
* $(-z^3)(5z) = -5z^4$
* $(-z^3)(z^2) = -z^5$
* $(-2z^2)(5z) = -10z^3$
* $(-2z^2)(z^2) = -2z^4$
* Combine like terms ($-5z^4$ and $-2z^4$ become $-7z^4$):
* Result: $-z^5 - 7z^4 - 10z^3$
* Standard form (highest power first): $-z^5 - 7z^4 - 10z^3$

4. What should we multiply with $(3x^2 + 2x)$ to get $(6x^3 + 4x^2)$?
* Look at the first term: To turn $3x^2$ into $6x^3$, you must multiply by $2x$ (because $3 \cdot 2 = 6$ and $x^2 \cdot x = x^3$).
* Check if this works for the second term: Does $2x \cdot 2x = 4x^2$? Yes.
* Answer: $2x$

5. Simplify each of the following expressions:

A. $(z^2y^4)^2$
* Apply the exponent of 2 to everything inside the parenthesis.
* $(z^2)^2 = z^{2 \cdot 2} = z^4$
* $(y^4)^2 = y^{4 \cdot 2} = y^8$
* Answer: $z^4y^8$

B. $(5p^2q^3)^3$
* Cube the number and multiply the exponents.
* $5^3 = 125$
* $(p^2)^3 = p^6$
* $(q^3)^3 = q^9$
* Answer: $125p^6q^9$

6. If $(3x^2 + 4y^2)3xy = ax^3y + 12xy^3$. Find the value of $a$.
* Expand the left side by distributing $3xy$.
* First part: $(3x^2)(3xy) = 9x^3y$
* Second part: $(4y^2)(3xy) = 12xy^3$
* So, the expanded form is $9x^3y + 12xy^3$.
* Compare this to the right side: $ax^3y + 12xy^3$.
* The coefficient of $x^3y$ is $a$ on the right and $9$ on the left.
* Answer: $a = 9$

7. If $3p(2q - 4r) = 6pq - mpr$. Find the value of $m$.
* Expand the left side:
* $3p \cdot 2q = 6pq$
* $3p \cdot (-4r) = -12pr$
* So, the equation is $6pq - 12pr = 6pq - mpr$.
* Compare the second terms: $-12pr$ matches $-mpr$.
* Therefore, $m = 12$.
* Answer: $m = 12$

8. If the height of a triangle is $(8 - 2b)$ and its base is $(b - 4)$, find the area of the triangle.
* Formula for Area of a Triangle: $\text{Area} = \frac{1}{2} \cdot \text{base} \cdot \text{height}$
* Substitute the values: $\text{Area} = \frac{1}{2}(b - 4)(8 - 2b)$
* First, multiply the binomials $(b - 4)(8 - 2b)$:
* $b \cdot 8 = 8b$
* $b \cdot (-2b) = -2b^2$
* $-4 \cdot 8 = -32$
* $-4 \cdot (-2b) = +8b$
* Combine: $-2b^2 + 16b - 32$
* Now, multiply by $\frac{1}{2}$:
* $\frac{1}{2}(-2b^2 + 16b - 32)$
* $-b^2 + 8b - 16$
* Answer: $-b^2 + 8b - 16$

Final Answer:
1. $12x^2$
2. $p^2x^4 + px^3 + p^2x^2$
3. A. $-12x^3 - 16x^2 - 6x - 8$
B. $-z^5 - 7z^4 - 10z^3$
4. $2x$
5. A. $z^4y^8$
B. $125p^6q^9$
6. $a = 9$
7. $m = 12$
8. $-b^2 + 8b - 16$