How do you simplify (x + 1)(5x^3 - x^2 + x - 4)?

May 17, 2017

Use the distributive property.
= $5 {x}^{4} + 4 {x}^{3} - 3 x - 4$

Explanation:

Using the distributive property, we would expand the brackets and condense the equation overall to make the simplified version. To do this in this case, this means multiplying every term (numbers seperated by plus signs or minus signs) in the first bracket by every term in the second one (see image below).

The equation would end up looking like this:
$= \left(x\right) \left(5 {x}^{3}\right) + \left(x\right) \left(- {x}^{2}\right) + \left(x\right) \left(x\right) + \left(x\right) \left(- 4\right) + \left(1\right) \left(5 {x}^{3}\right) + \left(1\right) \left(- {x}^{2}\right) + \left(1\right) \left(x\right) + \left(1\right) \left(- 4\right)$
You can skip writing the previous step down because you can go straight to the next step by doing the previous one in your head:
$= 5 {x}^{4} - {x}^{3} + {x}^{2} - 4 x + 5 {x}^{3} - {x}^{2} + x - 4$
And then we re-order it and group like terms (terms with the same variable name and exponent) to make it easy to condense.
$= 5 {x}^{4} - {x}^{3} + 5 {x}^{3} + {x}^{2} - {x}^{2} - 4 x + x - 4$
And then condense to its final form. (If you don't know how to add/subtract like terms, comment below).
$= 5 {x}^{4} + 4 {x}^{3} - 3 x - 4$

To verify our answer, we can sub a number into the variable $x$. Let's say 5:
$5 \cdot {5}^{4} + 4 \cdot {5}^{3} - 3 \cdot 5 - 4 = 3606$
$\left(5 + 1\right) \left(5 \cdot {5}^{3} - {5}^{2} + 5 - 4\right) = 3606$

Therefore, we did this correctly.