Distributive property is used to multiply polynomials. First let's start with just one number to distribute:
\(4 (x^2 - 3y)\)
Here, the \(4\) should be multiplied to every term within the brackets:
\((4)(x^2) + (4)(- 3y)\)
Notice we used an addition sign between the new terms. Remember your integer rules for multipliation (same sign = positive, opposite sign = negative).
\(4x^2 - 12y\)
To multiply polynomials, we need to multipy combinations of all the terms. You can use the acrynom FOIL to remember how to multiply two binomials. FOIL stands for First, Outer, Inner, Last.
Expand and simplify \((3-x^3)(4y+3) \).
First, we can create a table outlining the combinations and products of all terms using FOIL:
| F | O | I | L |
|---|---|---|---|
| \( (\colorbox{yellow}{3}-x^3)(\colorbox{yellow}{4y}+3) \) | \( (\colorbox{yellow}{3}-x^3)(4y+\colorbox{yellow}{3}) \) | \( (3\colorbox{yellow}{-x³})(\colorbox{yellow}{4y}+3) \) | \( (3\colorbox{yellow}{-x³})(4y+\colorbox{yellow}{3}) \) |
| \( 12y \) | \( 9 \) | \( -4x^3y \) | \( -3x^3 \) |
Then, we can add up the terms and collect like terms (if possible):
Another way to approach this problem is by splitting up the first polynomial and multiplying each term by the second polynomial:
\(= (3-x^3)(4y+3) \)
\(= (3)(4y+3) + (-x^3)(4y+3) \)
Next, we can distribute just one number:
\(= (12y)+(9) + (-4x^3y)+(-3x^3) \)
\(= 12y + 9 -4x^3y -3x^3 \)
We get the same answer as before.
Therefore, we can determine that the expression expanded and simplified is \(\boldsymbol{12y + 9 -4x^3y -3x^3} \).
\((2x-18)(3x+3) \)
First, we can use distributive property to multiple the polynomials. We can multiply the terms \(2x\) and \(-18\) by the second polynomial:
\(2x(3x + 3) - 18(3x + 3)\)
\((6x^2+6x) + (-54x-54)\)
Next, we can simplify by collecting like terms:
\(6x^2 + \textcolor{red}{6x} \textcolor{red}{-54x} - 54\)
\(6x^2-48x-54\)
Therefore, we can determine that the expression simplified is \(\boldsymbol{6x^2-48x-54}\).
\((2x+1)(3x^2−x+4) \)
First, we can use distributive property to multiply the polynomials. We can multiply the terms \(2x\) and \(1\) by the second polynomial:
\(2x \cdot (3x^2 - x +4) + 1\cdot(3x^2 - x +4)\)
\((6x^3-2x^2+8x)+(2x^2-x+4)\)
Next, we can simplify by collecting like terms:
\(6x^3\textcolor{red}{-2x^2} + \textcolor{red}{2x^2} + \textcolor{blue}{8x} \textcolor{blue}{-x} + 4\)
\(6x^3+x^2+7x+4\)
Therefore, we can determine that the expression simplified is \(\boldsymbol{6x^3+x^2+7x+4}\).