Adding and multiplying polynomials
If the area of a rectangle depends on $x$, adding areas or finding the area of a bigger rectangle means operating with polynomials. The rules are the same as with numbers; you just have to be careful to combine terms of the same kind.
Adding and subtracting polynomials
You add (or subtract) the like terms, that is, the terms with exactly the same variable part (same letters with the same exponents). In a polynomial in $x$, these are the terms of the same degree. It helps to line them up in columns, as in primary-school addition:
Given the polynomials $P(x) = 4x^2 - 3x + 5$ and $Q(x) = x^2 + 7x - 2$, perform the following operations by grouping like terms: a) $P(x) + Q(x)$ b) $P(x) - Q(x)$
Given the polynomials $P(x) = 4x^2 - 3x + 5$ and $Q(x) = x^2 + 7x - 2$, perform the following operations by grouping like terms:
- a) $P(x) + Q(x)$
- b) $P(x) - Q(x)$
Find the polynomial expressing the perimeter of the following geometric figure (an L-block from a video game). Remember that the perimeter is the sum of all its sides.
Find the polynomial expressing the perimeter of the following geometric figure (an L-block from a video game). Remember that the perimeter is the sum of all its sides.
A streamer charges a fixed amount of $50$ euros per livestream and, in addition, receives $2$ euros for each new subscription ($s$) and $0,5$ euros for each donation…
A streamer charges a fixed amount of $50$ euros per livestream and, in addition, receives $2$ euros for each new subscription ($s$) and $0,5$ euros for each donation ($d$).
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Write the algebraic expression (polynomial) representing their total income in one day.
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If in a livestream they get $20$ subscriptions and $40$ donations, how much money have they earned in total?
Multiplying polynomials
Monomial times monomial: you multiply the coefficients and the variable parts (adding the exponents of the same letter): $(3x^2)\cdot(-4x^3) = -12x^5$.
Polynomial times polynomial: each term of the first multiplies every term of the second (the distributive property). A rectangle shows this very well:
Perform the following multiplications of monomials, applying index laws: a) $(3x^2) \cdot (4x^3) =$ b) $(-2a^2b) \cdot (5ab^3) =$ c) $(\frac{1}{2}x) \cdot (6x^5) =$
Perform the following multiplications of monomials, applying index laws:
- a) $(3x^2) \cdot (4x^3) =$
- b) $(-2a^2b) \cdot (5ab^3) =$
- c) $(\frac{1}{2}x) \cdot (6x^5) =$
Apply the distributive property to calculate the area of the following rectangle, expressing the result as a polynomial:
Apply the distributive property to calculate the area of the following rectangle, expressing the result as a polynomial:
Apply the distributive property to multiply the following polynomials and collect like terms: a) $(2x - 3) \cdot (x^2 - 4x + 1)$ b) $(x^2 + 2) \cdot (x^2 - 2)$ c)…
Apply the distributive property to multiply the following polynomials and collect like terms:
- a) $(2x - 3) \cdot (x^2 - 4x + 1)$
- b) $(x^2 + 2) \cdot (x^2 - 2)$
- c) $-(x - 5) \cdot (2x + 3) + 2x^2$
Taking out a common factor
This is the distributive property in reverse: if all the terms have something in common, you take it outside, multiplying a bracket.
$$6x^3 - 9x^2 + 3x = 3x\,(2x^2 - 3x + 1)$$Factorise the following algebraic expressions by taking out the common factor to write them as the product of a monomial and a polynomial: a) $15x^4 - 10x^3 + 5x^2$ b)…
Factorise the following algebraic expressions by taking out the common factor to write them as the product of a monomial and a polynomial:
- a) $15x^4 - 10x^3 + 5x^2$
- b) $3a^2b - 6ab^2 + 9ab$
- c) $2x(x+1) - 3(x+1)$
Special products
There are three products that come up so often that it is worth knowing them by heart. The diagram explains where they come from:
Change $a$ and $b$ and see how the area of the big square is split up:
Calculate the resulting polynomial of the following expression using standard algebraic identities (square of a sum, square of a difference, and product of a sum and a…
Calculate the resulting polynomial of the following expression using standard algebraic identities (square of a sum, square of a difference, and product of a sum and a difference):
$A(x) = (2x + 3)^2 - (2x - 3)^2 + (x + 4) \cdot (x - 4)$
Simplify the final result as far as possible.
Determine the value of the coefficient $k$ so that the following equality between polynomials holds true: $(x + 3) \cdot (2x - 5) = 2x^2 + kx - 15$ Once $k$ is found,…
Determine the value of the coefficient $k$ so that the following equality between polynomials holds true:
$(x + 3) \cdot (2x - 5) = 2x^2 + kx - 15$
Once $k$ is found, work out the product $(x + 3) \cdot (2x - 5) \cdot (x + 1)$ and write the resulting polynomial in descending order of degree.