Sum and product of 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 things of the same kind.
Adding and subtracting polynomials
You add (or subtract) the like terms, that is, the ones with the same literal part (the same letters with the same exponents). In a polynomial in $x$, they are the ones of the same degree. It helps to line them up in columns, like in elementary 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 $1{,}000$ pesos per livestream and, in addition, receives $40$ pesos for each new subscription ($s$) and $10$ pesos for each…
A streamer charges a fixed amount of $1{,}000$ pesos per livestream and, in addition, receives $40$ pesos for each new subscription ($s$) and $10$ pesos 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): $(3x^2)\times(-4x^3) = -12x^5$.
Polynomial times polynomial: each term of the first multiplies every term of the second (distributive property). A rectangle shows this very well:
Perform the following multiplications of monomials, applying index laws: a) $(3x^2) \times (4x^3) =$ b) $(-2a^2b) \times (5ab^3) =$ c) $(\frac{1}{2}x) \times (6x^5) =$
Perform the following multiplications of monomials, applying index laws:
- a) $(3x^2) \times (4x^3) =$
- b) $(-2a^2b) \times (5ab^3) =$
- c) $(\frac{1}{2}x) \times (6x^5) =$
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) $(2x - 3) \times (x^2 - 4x + 1)$ b) $(x^2 + 2) \times (x^2 - 2)$ c)…
Apply the distributive property to multiply the following polynomials and collect like terms:
- a) $(2x - 3) \times (x^2 - 4x + 1)$
- b) $(x^2 + 2) \times (x^2 - 2)$
- c) $-(x - 5) \times (2x + 3) + 2x^2$
Factoring out a common factor
It is the distributive property in reverse: if all the terms have something in common, you take it out and multiply it by a set of parentheses.
$$6x^3 - 9x^2 + 3x = 3x\,(2x^2 - 3x + 1)$$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
Some products show up so often that it's worth knowing them by heart: they are the special products. They are the basis of the factoring you will use to solve quadratic equations. Their diagrams explain where they come from:
The result of squaring a binomial is called a perfect square trinomial, and the result of multiplying conjugate binomials is called a difference of squares.
Change $a$ and $b$ and watch how the area of the big square is divided 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):
$A(x) = (2x + 3)^2 - (2x - 3)^2 + (x + 4) \times (x - 4)$
Simplify the final result as far as possible.
Work out the following combined operation and simplify the result as far as possible, ordering the polynomial in descending order of degree: $$(2x^2 - 3) \times (x^2 +…
Work out the following combined operation and simplify the result as far as possible, ordering the polynomial in descending order of degree:
$$(2x^2 - 3) \times (x^2 + 5x - 1) - [(x^2 - 2)^2 + 6x^3]$$Pay special attention to expanding the algebraic identity and the negative sign affecting the square bracket.
Binomials with a common term
When the two binomials share the same term ($x$) and only the number changes, the product comes out directly: $x^2$, plus the sum of the numbers times $x$, plus their product.
Determine the value of the coefficient $k$ so that the following equality between polynomials holds true: $(x + 3) \times (2x - 5) = 2x^2 + kx - 15$ Once $k$ is found,…
Determine the value of the coefficient $k$ so that the following equality between polynomials holds true:
$(x + 3) \times (2x - 5) = 2x^2 + kx - 15$
Once $k$ is found, work out the product $(x + 3) \times (2x - 5) \times (x + 1)$ and write the resulting polynomial in descending order of degree.
A courier company uses boxes whose dimensions (in cm) are in arithmetic progression and depend on a variable $x$: Width: $x - 2$ Length: $x + 1$ Height: $x + 4$ Find the…
A courier company uses boxes whose dimensions (in cm) are in arithmetic progression and depend on a variable $x$:
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Width: $x - 2$
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Length: $x + 1$
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Height: $x + 4$
Find the polynomial $V(x)$ expressing the volume of the box. Then, find the polynomial $S(x)$ representing the total surface area of the six faces of the box. Simplify both expressions.