Simplify: r^4+r^3-7r^2-r+6=0 Tiger Algebra Solver (2024)

2.1 Find roots (zeroes) of : F(r) = r⁴+r³-7r²-r+6
Polynomial Roots Calculator is a set of methods aimed at finding values ofrfor which F(r)=0

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers r which can be expressed as the quotient of two integers

The Rational Root Theorem states that if a polynomial zeroes for a rational numberP/Q then P is a factor of the Trailing Constant and Q is a factor of the Leading Coefficient

In this case, the Leading Coefficient is 1 and the Trailing Constant is 6. The factor(s) are:

of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2 ,3 ,6 Let us test ....

The Factor Theorem states that if P/Q is root of a polynomial then this polynomial can be divided by q*x-p Note that q and p originate from P/Q reduced to its lowest terms

In our case this means that
r⁴+r³-7r²-r+6
can be divided by 4 different polynomials,including by r-2

2.3 Find roots (zeroes) of : F(r) = r³+3r²-r-3

See theory in step 2.1
In this case, the Leading Coefficient is 1 and the Trailing Constant is -3. The factor(s) are:

of the Leading Coefficient : 1
of the Trailing Constant : 1 ,3 Let us test ....

The Factor Theorem states that if P/Q is root of a polynomial then this polynomial can be divided by q*x-p Note that q and p originate from P/Q reduced to its lowest terms

In our case this means that
r³+3r²-r-3
can be divided by 3 different polynomials,including by r-1

2.5Factoring r²+4r+3

The first term is, r² its coefficient is 1.
The middle term is, +4r its coefficient is 4.
The last term, "the constant", is +3

Step-1 : Multiply the coefficient of the first term by the constant 1•3=3

Step-2 : Find two factors of 3 whose sum equals the coefficient of the middle term, which is 4.

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step2above, 1 and 3
r² + 1r+3r + 3

Step-4 : Add up the first 2 terms, pulling out like factors:
r•(r+1)
Add up the last 2 terms, pulling out common factors:
3•(r+1)
Step-5:Add up the four terms of step4:
(r+3)•(r+1)
Which is the desired factorization

Root plot for : y = r²+4r+3
Axis of Symmetry (dashed) {r}={-2.00}
Vertex at {r,y} = {-2.00,-1.00}
r-Intercepts (Roots) :
Root 1 at {r,y} = {-3.00, 0.00}
Root 2 at {r,y} = {-1.00, 0.00}

4.2Solvingr²+4r+3 = 0 by Completing The Square.Subtract 3 from both side of the equation :
r²+4r = -3

Now the clever bit: Take the coefficient of r, which is 4, divide by two, giving 2, and finally square it giving 4

Add 4 to both sides of the equation :
On the right hand side we have:
-3+4or, (-3/1)+(4/1)
The common denominator of the two fractions is 1Adding (-3/1)+(4/1) gives 1/1
So adding to both sides we finally get:
r²+4r+4 = 1

Adding 4 has completed the left hand side into a perfect square :
r²+4r+4=
(r+2)•(r+2)=
(r+2)²
Things which are equal to the same thing are also equal to one another. Since
r²+4r+4 = 1 and
r²+4r+4 = (r+2)²
then, according to the law of transitivity,
(r+2)² = 1

We'll refer to this Equation as Eq. #4.2.1

The Square Root Principle says that When two things are equal, their square roots are equal.

Note that the square root of
(r+2)² is
(r+2)^2/2=
(r+2)¹=
r+2

Now, applying the Square Root Principle to Eq.#4.2.1 we get:
r+2= √ 1

Subtract 2 from both sides to obtain:
r = -2 + √ 1

Since a square root has two values, one positive and the other negative
r² + 4r + 3 = 0
has two solutions:
r = -2 + √ 1
or
r = -2 - √ 1

4.3Solvingr²+4r+3 = 0 by the Quadratic Formula.According to the Quadratic Formula,r, the solution forAr²+Br+C= 0 , where A, B and C are numbers, often called coefficients, is given by :

-B± √B²-4AC
r = ————————
2A In our case,A= 1
B= 4
C= 3 Accordingly,B²-4AC=
16 - 12 =
4Applying the quadratic formula :

-4 ± √ 4
r=—————
2Can √ 4 be simplified ?

Yes!The prime factorization of 4is
2•2
To be able to remove something from under the radical, there have to be 2 instances of it (because we are taking a square i.e. second root).

√ 4 =√2•2 =
±2 •√ 1 =
±2

So now we are looking at:
r=(-4±2)/2

Two real solutions:

r =(-4+√4)/2=-2+= -1.000

or:

r =(-4-√4)/2=-2-= -3.000