Question

Polynomials??

Would someone please explain polynomials to me? I'm in 8th grade Algebra 1 and I don't understand them at all. Thanks!

Answer 1

`"See explanation"`

Explanation:

`"I see you started only algebra so this will be a little too"`
`"complicated. I refer to the other answer for general"`
`"polynomials in several variables."`

`"I gave the theory for polynomials in one variable x."`

`"A polynomial in one variable x is a sum of integer powers of"`
`"that variable x, with a number, named the coefficient, in front"`
`"of each power term."`
`"We arrange the power terms from left to right, with the higher"`
`"power terms first, so in descending order :"`

`y = f(x) = x^2 + 3 x - 4 , " example given."`

`"The degree of the polynomial is the exponent of the highest"`
`"power, so the example is a polynomial of degree 2."`
`"When we put the polynomial equal to zero, we have a"`
`"polynomial equation."`

`x^2 + 3 x - 4 = 0 , " is a quadratic equation example given."`

`"If the degree is 1 we call it a linear equation."`
`"If the degree is 2 we call it a quadratic equation."`
`"If the degree is 3 we call it a cubic equation."`
`"And so on : quartic (degree 4), quintic, sextic, septic, ..."`

`5 x + 6 = 0 ,`

`"is a linear equation, we solve it by doing"`

`=> 5 x = -6" (subtracting 6 on both sides of the equation)"`
`=> x = -6/5" (dividing both sides of the equation by 5)"`

`"This is correct as you see that, when we plug in the value"`
`"-6/5 for x, we get zero."`

`"We say that -6/5 is the solution or the zero or the root of that"`
`"equation."`

`"Now if you did not learn about quadratic equation yet, you"`
`"do not have to read further."`

`"Now most examples are quadratic equations because the"`
`"ones with degree higher than 2 are generally difficult to"`
`"solve."`

`"One solving method for a quadratic equation is completing"`
`"the square :"`

`x^2 + 3 x - 4 = (x + 1.5)^2 - 6.25 = 0`

`"(because (x+a)² = x² + 2a x + a²)"`

`=> (x + 1.5)^2 = 6.25`
`=> x + 1.5 = pm 2.5`
`=> x = -1.5 pm 2.5`
`=> x = -4 or 1`

`"Another solving method for quadratic equations is the formula"`
`"with the discriminant : "`

`x = (-b pm sqrt(b^2-4ac))/(2a)`

`"for " a x^2 + b x + c = 0`
`"Here in the example we have : "a = 1, b = 3, c = -4."`
`"So we plug this in the formula and get"`

`x = (-3 pm sqrt(3^2-4*1*(-4)))/(2*1)`
`= (-3 pm sqrt(9+16))/2`
`= (-3 pm sqrt(25))/2`
`= (-3 pm 5)/2`
`= -4 or 1`

`"Another solving method for polynomial equations in general"`
`"is factoring."`

`x^3 + 3 x^2 + x + 3 = 0`
`=> (x^3 + x) + (3 x^2 + 3) = 0`
`=> x(x^2+1) + 3(x^2+1) = 0`
`=> (x^2+1)(x+3) = 0`
`=> x=-3 " ("x^2+1 > 0, " so here we have only 1 real root)"`

`"If a is a root, (x-a) is a factor."`
`"And a polynomial equation of degree n has at most n real roots."`

Answer 2

A polynomial has 'many' terms. `" "4x^3-2xy +2x+3`

Explanation:

In algebra we call maths sentences expressions.

An expression is made up of terms, which can have numbers and letters (called variables).

An English sentence is made up of words. (like this one)
A Maths expression is made up of terms.

Terms are separated from each other by `+ and -` signs.

`3x^4 - 5x^3 +4x^2 -7x+11" "` has `" "5` terms

If there is only one term, it is called a monomial: `" "5xy^2`

If there are two terms, it is called a bionomial: `" "2x -3y`

If there are three terms, it is called a trinomial: `" "2x -3y +5`

The prefix 'poly' means 'many.
(Many means 2 or more, but we usually have 4 or more terms)

So a polynomial has 'many' terms. `" "4x^3-2xy +2x+3`

There are other restrictions for defining a polynomial, but in Grade 8, you do not need to know them yet.

At this stage you will be learning to do the different operations in algebra using expressions, (or polynomials)

You need to know that you can only add or subtract if you have 'like terms' which means that the variable parts are exactly the same.

`3xy +7xy -2xy = 8xy`

However, you can multiply or divide any terms.

`3xy^2 xx 4x^2yz = 12x^3y^3z`