When combining like terms, you must look at the variables and coefficients of the terms and whether they are added or subtracted.
Like terms can be expressed as products of the same variable multiplied by a common coefficient. For example, 2x × 2x is a like term, since they both have an x as the variable and 2 as the coefficient.
When adding or subtracting like terms, you must decide if the new term is higher or lower in magnitude (exponent) than other like terms. If it is of higher magnitude, then you must add zeroes to the original number to make them match. If it is of lower magnitude, then you must find a matching one and add zeroes to make them match in magnitude.
Using these rules when combining like terms will help you simplify expressions more efficiently.
7×3 – 4×2 + 2×3 – 4×2 = 9×3
When you add polynomials, you must make sure they are of the same degree. If one polynomial has a higher degree than the other, you must find a polynomial of the same degree as the sum to add it.
For example, if you are trying to add a quadratic and a linear polynomial, you must find a linear polynomial to add to the quadratic. You could not just add them because that would not make sense mathematically.
Making sure the degrees are matched up correctly is important because if you do not, then your answer will not make sense. For example, trying to add a linear and a quadratic polynomial will result in a cubic polynomial due to having more terms in the second polynomial.
Check your work when adding polynomials to make sure your answer is of the correct degree.
Divide by the largest exponent
When you add polynomials, you have to make sure that the coefficients and the variable values are the same. If they are not the same, you have to find a common denominator or radical that they are all in relation to.
Then, you can add the terms together according to their respective exponents. For example, if you had (a + b) + (a – b), you would add the coefficients and bases and then subtract one base from another according to their respective exponents.
The hardest part about adding polynomials is when one of the terms has an exponent that is higher than the other term(s) by one. In this case, you have to divide one of the terms by an exponent equal to or less than that of the other term.
For example, if you had (+b) + (–b), you would have to divide –b by –1 in order to make both exponents -1.
9×3 ÷ 3 = 3(1 + x + x2)
When factoring polynomials, you may need to find the sum of the corresponding monomial factors. This is called partial factoring.
For example, let’s look at the polynomial 9×3 ÷ 3. We can’t divide by 3 until we find its corresponding fractional coefficient.
So let’s assume there is a 1 hidden in the middle. We can now divide both sides by 3 and 1 to get our final answer: 1 + x + x2.
Partial factoring is important to know because it can help you find hidden monomial coefficients when dividing polynomials. It also helps when finding sums of polynomials later on.
This is one possible answer, but not the only one
When adding polynomials, you can add the coefficients and then add the roots of the coefficients. For example, 2×2 + 2×2 would be 4×2.
In the given problem, the first polynomial has a coefficient of 7 and a root of 3, while the second polynomial has a coefficient of 2 and a root of 4. Adding these gives you 7 + 4 = 11, which, when factored, would be (3+1)×2=6×2=12.
The answer could have also been (6×2)+(12×3)=48+36=84. These are both possible answers, but one is not necessarily more correct than the other. It depends on what the question asks for.
Try another example, such as (8y4 – 4y2) + (4y4 – 8y2)
In this case, the coefficients of the terms are reversed, so the red term has a higher coefficient. Since the red term has the higher coefficient, it will be larger in magnitude.
When you add polynomials, you have to make sure you are adding all of the same kind of term. If you were to add these two polynomials, you would have to make sure both had y2 as a coefficient in one of the terms or you would get an error.
You can always cancel out coefficients to make things easier. In this case, you could cancel out the y2 from the second polynomial and still have the correct answer.
See if you can come up with a general method for solving these types of polynomial equations
So now that we’ve covered how to add and subtract polynomials, let’s talk about how to multiply them.
Like with adding and subtracting polynomials, there are two ways to multiply polynomials: using the distributive property and using the product rule. We’ll cover both here!
The distributive property is a pretty simple concept. It states that when you multiply a sum of things by a factor, the overall value remains the same. For example, if you have 2 + 2 = 4, then 2 × 4 = 8. The overall value is 8, even though you had to distribute the 2 over the 8.