Curriculum Outcomes
Algebra and Number: 4. Demonstrate an understanding of the multiplication of polynomial expressions (limited to monomials, binomials and trinomials), concretely, pictorially and symbolically. [CN, R, V] 5. Demonstrate an understanding of common factors and trinomial factoring, concretely, pictorially and symbolically. [C, CN, R, V] |
Things You Need to Know
- Monomials are a product of a bunch of terms. (examples: 3, x, 8x, 24xy)
- Polynomials are a sum of a bunch of monomials. (examples: (2x + 4), (x + 15y - 23), (12xy + 45x - 14y + 5))
- Binomials are polynomials where the sum is of only two monomials. (examples: (20x + 40), (15y + 5x))
- Trinomials are polynomials where the sum is of three monomials. (examples: (3x + 8y - 4), (4a - 6b + 19ab))
- When multiplying polynomials, we must make sure that each monomial of the first polynomial is multiplied by each monomial of the second polynomial (this is called the "distributive property"). Then we must combine all the "like terms" (monomials with both the same variable(s) and with the same exponent(s)).
- Example: (x + 3)(x - 2) = x² - 2x + 3x - 6
- Then combine like terms: = x² + 1x - 6
- The greatest common factor (GCF) of two or more numbers is the largest number that is a factor of all those numbers.
- Example: the GCF of 12 and 15 = 3.
- The GCF of x^3 and x² = x²
- You can use the GCF to factor a polynomial.
- Find the GCF of all the monomials in the polynomial. To do this, find the GCF of the coefficients (the whole numbers), and the GCF of each of the variables.
- Put the GCF on the outside of the brackets.
- For each monomial, identify what the GCF would have to multiply by to get that monomial.
- Example: Factor (2x² + 6x + 10xy). The GCF of all the monomials is 2x. So we have 2x(x + 3 + 5y).
- You can also factor by grouping, which gives a binomial as a GCF instead of a monomial.
- Example: Use factoring by grouping to factor: (10x + 5y + 2mx + my)
- Look at the first two terms separately, and the last two terms separately.
- First two: the GCF of 10x and 5y = 5. So we have 5(2x + y).
- Last two: the GCF of 2mx and my = m. So we have m(2x + y).
- The first two and last two both share (2x + y). We can use this as a binomial, and the GCF of first two + GCF of last two as the second binomial.
- So we have a final answer of: (2x + y)(5 + m).
- If we have a trinomial of the form x² + bx + c (where b and c are integers), we'd ideally like to factor it into two binomials.
- First, find two integers (we'll call them p and q) such that they add together to equal b, and they multiply together to equal c.
- In other words, p + q = b, and pq = c. We've called this the SP rule (Sum/Product Rule).
- Second, you can rewrite this trinomial as a product of two binomials: (x + p)(x + q)
- Note: if c is positive, then we know p and q are either both positive or both negative. If c is negative, p and q are opposite signs (one positive one negative).
- If we have a trinomial of the form ax² + bx + c (where a, b, and c are integers), we follow these steps to factor it:
- Factor out the GCF of all the monomials if possible.
- Find two numbers (we'll call them p and q) with a product of (a x c) and a sum of b (using the a, b, and c values in your GCF-factored trinomial, not necessarily the original trinomial).
- In other words, (pq = ac) and (p + q = b).
- Next, rewrite your GCF-factored trinomial like so: (ax² + px + qx + c)
- Now, factor by grouping.
- A special type of polynomial we can factor is called the "difference of squares": (x² - y²)
- If we have a polynomial of this form, where x or y may be known, we can factor it like this:
- (x² - y²) = (x + y)(x - y)
- Example: Factor (x² - 64). We notice that 64 is a perfect square (8 x 8 = 64). So this is a difference of squares. We can factor it using the difference of squares rule: (x + 8)(x - 8).