Solving polynomial equations  Operations with Polynomials
What is the zero product property
The zero product property definition tells us that if:then either a or b has to equal 0.
Logically, this makes sense. Any number multiplied by 0 gives us 0. No other combinations can give us this. Therefore, one of the two unknown numbers must be 0.
How to solve polynomials
When weβre asked to solve a polynomial, weβre actually being asked to find the roots where the function equals 0. You can see how the above zero product property can come in handy! So how do we start solving polynomials? Weβll have to do some isolating.
How to isolate a variable
In order to figure out when a polynomial equals 0, weβll have to isolate the variable that is in the equation so that we can find its value. This is like how we solve twostep linear equations. Letβs try it out with a simple polynomial equation. Do you know how to solve for x?
Since we donβt know what x is, weβll have to isolate it onto one side to see what it equals. In this case, letβs start with moving the 1, and then the 2 thatβs next to the x.
When youβve got a positive on one side and youβve got to move it to the other side of the equal sign, simply reverse the sign to negative. If itβs negative and youβve got to move it, itβll become positive. Similarly, if you have to multiply on one side and now youβre trying to move it to the other side, youβll have to divide it. If youβve got a division, itβll become multiplication on the other side of the equal sign.
If youβve got a variable on both sides of the equal sign, youβll still have to group it to one side in order to find its value.
Example problems
Question 1:
1a) Solve the equation:
$4x^{2}  9 = 0$
Solution:
Since weβre dealing with a degree of a polynomial (in this case itβs the power of 2), when you move a square to the other side of the equal sign, youβll have a square root. A note is that all positive real numbers has two square rootsβone thatβs positive and one thatβs negative. Therefore, weβll put a Β±sign in front of the square root to indicate weβd like both the negative and positive in our final answer.
$4x^{2} = 9$
$x^{2} = \frac{9}{4}$
$x = \pm \sqrt{\frac{9}{4}}$
$x = \pm \frac{3}{2}$
1b) Solve the equation:
$(2x  3)^{2} = 1$
Solution:
$2x  3 = \pm \sqrt{1}$
$2x  3 = \pm 1$
$2x = \pm 1 + 3$
$x = \frac{3 \pm 1}{2}$
$x = 2, 1$
Question 2
Solving the equation:
$3x^{2}  13x  10 = 0$
We have to factor the polynomial before solving for x so that we can get it into a form that looks like weβre about to do distributive property on it.
After factoring, we have:
$(3x + 2)(x  5) = 0$
To solve for x, we can separate $(3x + 2)(x  5) = 0$ into two equations using the zero product property:
$(3x + 2) = 0$,and $(x  5) = 0$
Isolate and solve for x
$3x + 2 = 0$
$3x = 2$
$x = \frac{2}{3}$
$x  5 = 0$
$x = 5$
So, in this question, $x = \frac{2}{3}$ and $x = 5$
Not sure about your answers? Hereβs an online polynomial equation calculator you can use to check your answer when solving polynomial equations.
Want to move on? You can take a look at these related lessons to improve on your understanding:
 Determining the number of solutions to a linear equations
 Solving systems of linear equations by graphing
Solving polynomial equations
Lessons

a)
What is zero product property?


1.
Solving the equations: