Summary
Highlights
The video begins by explaining how to add polynomials. The core principle is to identify and combine like terms, which are terms with the same variable and exponent. The numerical coefficients are added, while the common variable is retained. Examples include adding 3y + 2y to get 5y, and 3u³ + (-9u³) resulting in -6u³. For more complex expressions like (2a - 1) + (4a + 3), it's recommended to rearrange terms so like variables are grouped together, making addition simpler.
Subtracting polynomials follows similar rules to addition, but with an additional initial step. The method involves copying the minuend, changing the operation to addition, and then taking the opposite sign of the subtrahend. For instance, in 2w - 5w, it becomes 2w + (-5w), leading to -3w. Another example shows -4x²y - 2x²y becoming -4x²y + (-2x²y) which simplifies to -6x²y. When subtracting binomials like (2a + 4b) - (5a + 3b), the signs inside the second parenthesis are flipped to (2a + 4b) + (-5a - 3b), and then like terms are combined.
Multiplying polynomials requires a good understanding of the laws of exponents. For multiplying monomials, such as -3m²n³ * 5mn², you multiply the numerical coefficients and then multiply the literal coefficients by adding their exponents. This results in -15m³n⁵. When multiplying a monomial by a trinomial, like 2x * (x² - 4x + 3), the distributive property is used. Each term in the trinomial is multiplied by the monomial, leading to 2x³ - 8x² + 6x. For binomials multiplied by binomials, the FOIL (First, Outer, Inner, Last) method is introduced. An example, (2w + 3) * (5w + 4), demonstrates how to apply FOIL to get 10w² + 8w + 15w + 12, which simplifies to 10w² + 23w + 12 after combining like terms.
Dividing polynomials is explained through several examples, starting with dividing monomials. To divide 12m⁷ by 4m³, the expression is written as a fraction, numerical coefficients are divided (12/4 = 3), and exponents of like variables are subtracted (7-3 = 4), resulting in 3m⁴. When dividing a polynomial by a monomial, such as (18m - 24) / 2, each term of the polynomial is divided separately by the monomial (18m/2 - 24/2), yielding 9m - 12. For more complex polynomial division, like (3a⁵ - 5a³ + 2a²) / a², each term in the numerator is divided by a², applying the exponent subtraction rule, which simplifies to 3a³ - 5a + 2 (since any base raised to the power of zero equals 1).