What is a quadratic polynomial?

Also, if the two factors are the same, then there are still two solutions.

(x - 4)^2 = 0

The solutions are 4 and 4.

When we count solutions like that, we're considering something called "multiplicity of roots". (That's a fancy way of saying, "repeated roots".)

Repeated roots manifest themselves graphically in a special way. That is, when we look at a polynomial's graph, we can tell if there is any repeated root within the polynomial's factorization.

:)
 
eddy2017 said:
Should I understand by this that the x^2 can also be the second term, that not necessarily have to be at the beginning?.
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No, that's not the point, Eddy. (After all, the Commutative Property tells us that we may write polynomial terms in any order we like.)

Jomo would like you to recognize a quadratic form, in that fourth-degree polynomial. How about you rewrite the polynomial, after making a u-substitution. You will then see the quadratic form.

Let u = x^2

Then u^2 is (x^2)^2 or x^4.

:)
 
Otis said:
No, that's not the point, Eddy. (After all, the Commutative Property tells us that we may write polynomial terms in any order we like.)

Jomo would like you to recognize a quadratic form, in that fourth-degree polynomial. How about you rewrite the polynomial, after making a u-substitution. Perhaps, you will then see the quadratic form.

Let u = x^2

Then u^2 is (x^2)^2 or x^4.

:)
Click to expand...
Well, that is good then. I thought the x^2 (or variation of it like x^4) needed to be the first term. So this is clear now. Could be in any position.
X^4= x^2 x^2
To be honest that is what I had thought.
Thank you Otis for the explanation.
so then it could also be x^8 because x^2 * x^2* X^2 =x^8 , right?
 
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And another question :
Does it always have to boil down to x^2?.
It can never be x^3, right?.
 
Eddy

This a technical point. You need to distinguish between expressions and equations.

An expression in elementary algebra represents a number (although the number may be unknown or unspecified). [imath]x^2 + 2[/imath] is a quadratic expression.

An equation in elementary algebra is a statement that two expressions represent the same number. [imath]x^2 + 2 = 3x[/imath] is an equation.

Something is in canonical form if it is in a form that is quite useful and very commonly used. An equation may be quadratic even though it is not in canonical form. A canonical form for quadratic equations in one variable is [imath]ax^2 + bx + c = 0.[/imath]
 
JeffM said:
Eddy

This a technical point. You need to distinguish between expressions and equations.

An expression in elementary algebra represents a number (although the number may be unknown or unspecified). [imath]x^2 + 2[/imath] is a quadratic expression.

An equation in elementary algebra is a statement that two expressions represent the same number. [imath]x^2 + 2 = 3x[/imath] is an equation.

Something is in canonical form if it is in a form that is quite useful and very commonly used. An equation may be quadratic even though it is not in canonical form. A canonical form for quadratic equations in one variable is [imath]ax^2 + bx + c = 0.[/imath]
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I understand now. Thank you, Prof.
 
Now I can help explain about u-substitutions. Sometimes we have an expression or equation that APPEARS very hard to work with. There may, however, be a substitution that will turn the expression or equation into something that is easy to work with. Example

[math]x^6 + 216 = 35x^3.[/math]
That is a polynomial equation in the sixth degree in one variable. In general, the solution to such equations is difficult; there is not necessarily a solution in radicals.. However, if we "see" the substitution

[math]\text {Let } u = x^3 \implies u^2 + 216 = 35u \implies u^2 - 35u + 216 = 0.[/math]
Now that is a quadratic equation in canonical form, and that is easy to solve.

[math]\therefore u = 8 \text { or } 27 \implies x^3 = 8 \text { or } 27 \implies x = 2 \text { or } 3.[/math]
Let's check our answers.

[math]2^6 + 216 = 64 + 216 = 280 = 35 * 8 = 35 * 2^3. \ \checkmark\\ 3^6 + 216 = 729 + 216 = 945 = 35 * 27 = 35 * 3^3.\ \checkmark[/math]
One of the reasons that we study so intently relatively simple types of equations is so we can reduce more complex ones into things we can work with. But I must admit that u-substitutions are art rather than science. They take a ton of experience so don't stress about them right now. They are generally taught about the time you transition from algebra to calculus.

PS I am not a professor.
 
JeffM said:
Now I can help explain about u-substitutions. Sometimes we have an expression or equation that APPEARS very hard to work with. There may, however, be a substitution that will turn the expression or equation into something that is easy to work with. Example

[math]x^6 + 216 = 35x^3.[/math]
That is a polynomial equation in the sixth degree in one variable. In general, the solution to such equations is difficult; there is not necessarily a solution in radicals.. However, if we "see" the substitution

[math]\text {Let } u = x^3 \implies u^2 + 216 = 35u \implies u^2 - 35u + 216 = 0.[/math]
Now that is a quadratic equation in canonical form, and that is easy to solve.

[math]\therefore u = 8 \text { or } 27 \implies x^3 = 8 \text { or } 27 \implies x = 2 \text { or } 3.[/math]
Let's check our answers.

[math]2^6 + 216 = 64 + 216 = 280 = 35 * 8 = 35 * 2^3. \ \checkmark\\ 3^6 + 216 = 729 + 216 = 945 = 35 * 27 = 35 * 3^3.\ \checkmark[/math]
One of the reasons that we study so intently relatively simple types of equations is so we can reduce more complex ones into things we can work with. But I must admit that u-substitutions are art rather than science. They take a ton of experience so don't stress about them right now. They are generally taught about the time you transition from algebra to calculus.

PS I am not a professor.
Click to expand...
Amazing. Thanks a lot, JeffM. I'm printing and studying all this and filing it for future reference.
 
eddy2017 said:
And another question :
Does it always have to boil down to x^2?
Click to expand...
Again, I don't know what you have in mind.

What is 'it'?

Please try to stop using lots of unreferenced pronouns. I cannot understand that kind of English.

:(
 
Otis said:
Again, I don't know what you have in mind.

What is 'it'?

Please try to stop using lots of unreferenced pronouns. I cannot understand that kind of English.

:(
Click to expand...
Sorry, what I tried to mean was that if in a quadratic equation the exponent of the x has to be a perfect square?.
If I could find this x^7 in a quadratic equation, or does it always have to be x to the two
Or x to the fourth
x to the eight...
 
eddy2017 said:
And another question :
Does it always have to boil down to x^2?.
It can never be x^3, right?.
Click to expand...
No, that is the point that I made in my post where I said that 3x^4 + 2x^2 - 11 = 0 is a quadratic equation in x^2.

x3 + 7x3/2 + 13 = 0 can be thought of as a quadratic form! How can this be??
 
eddy2017 said:
Sorry, what I tried to mean was that if in a quadratic equation the exponent of the x has to be a perfect square?.
If I could find this x^7 in a quadratic equation, or does it always have to be x to the two
Or x to the fourth
x to the eight...
Click to expand...
No, the exponent does not have to be a perfect square. JeffM showed you an example where the powers of x are 6, 3 and 0. Neither 6 nor 3 is a perfect square.
My example above has the powers of x being 3, 3/2 and 0.
 
eddy2017 said:
what I tried to mean ...
in a quadratic equation the exponent of the x has to be a perfect square?
Click to expand...
Not at all, Eddy. All quadratic equations in one variable may be put into the form:

ax^2 + bx + c = 0

All polynomial exponents on variables must be positive Integers. That's the only requirement for the exponents. The exponents do not need to be perfect squares.

One of your books must provide a definition for polynomials. Can you try to locate that definition?

:)
 
eddy2017 said:
I could find this x^7 in a quadratic equation
Click to expand...
No, you can not, but you might find it in a quadratic form.

Quadratic equations contain quadratic polynomials. All quadratic polynomials in one variable can be written as ax^2+bx+c.

There will never be any exponent larger than 2, in a quadratic polynomial. That's the definition! Quadratic polynomials are second-degree polynomials.

Maybe you are conflating the names 'quadratic polynomial' and 'quadratic form' and 'quadratic equation'. They do not mean the same thing.


Some polynomials containing an x^7 term may be expressed in quadratic form, but only by changing the inputs (eg: u-substitution). Whenever we do that, we are switching to a new function. We are not changing the original polynomial into a quadratic; we are simply using a quadratic form to help us obtain information about the original, higher-order polynomial.

Did you try the u-substitution example, that I'd suggested. If so, then what did you get?

:)
 
Otis said:
Just to be clear for readers, the left-hand side above is not a polynomial.

:)
Click to expand...
... and that is the interesting part! The lhs is not a polynomial, but it can be thought of as a polynomial with the correct u-substitution.
 
Otis said:
Not at all, Eddy. All quadratic equations may be put into the form:

ax^2 + bx + c = 0

All polynomial exponents must be positive Integers. That's the only requirement for the exponents. They do not need to be perfect squares.

One of your books must provide a definition for polynomials. Can you try to locate that definition?

:)
Click to expand...
Algebraic expression that constitutes the ordered sum or subtraction of a finite number of terms or monomials. They can have more than one variable.
 
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