4. Factorise : x3 + x2 - 26x + 24
I have underlined that I don't understand how it came so plz can anyone slove this and explain me how it came
Answers
Answer:
add and subtract x2 in whole equation
x3+x2-26x+24
= x3+x2-26x+24+x2-x2
= x3-x2+2x2-26x+24
= now write 26x into (-2x-24x)
that's how this equation come
= x3-x2+2x2-2x-24x+24
hope this will help u
Reformatting the input :
Changes made to your input should not affect the solution:
(1): "x2" was replaced by "x^2". 1 more similar replacement(s).
Step by step solution :STEP1:Checking for a perfect cube
1.1 x3-x2-26x-24 is not a perfect cube
Trying to factor by pulling out :
1.2 Factoring: x3-x2-26x-24
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: -26x-24
Group 2: x3-x2
Pull out from each group separately :
Group 1: (13x+12) • (-2)
Group 2: (x-1) • (x2)
Bad news !! Factoring by pulling out fails :
The groups have no common factor and can not be added up to form a multiplication.
Polynomial Roots Calculator :
1.3 Find roots (zeroes) of : F(x) = x3-x2-26x-24
Polynomial Roots Calculator is a set of methods aimed at finding values of x for which F(x)=0
Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers x which can be expressed as the quotient of two integers
The Rational Root Theorem states that if a polynomial zeroes for a rational number P/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 -24.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2 ,3 ,4 ,6 ,8 ,12 ,24
Let us test ....
P Q P/Q F(P/Q) Divisor -1 1 -1.00 0.00 x+1 -2 1 -2.00 16.00 -3 1 -3.00 18.00 -4 1 -4.00 0.00 x+4 -6 1 -6.00 -120.00 -8 1 -8.00 -392.00 -12 1 -12.00 -1584.00 -24 1 -24.00 -13800.00 1 1 1.00 -50.00 2 1 2.00 -72.00 3 1 3.00 -84.00 4 1 4.00 -80.00 6 1 6.00 0.00 x-6 8 1 8.00 216.00 12 1 12.00 1248.00 24 1 24.00 12600.00
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
x3-x2-26x-24
can be divided by 3 different polynomials,including by x-6
Polynomial Long Division :
1.4 Polynomial Long Division
Dividing : x3-x2-26x-24
("Dividend")
By : x-6 ("Divisor")
dividend x3 - x2 - 26x - 24 - divisor * x2 x3 - 6x2 remainder 5x2 - 26x - 24 - divisor * 5x1 5x2 - 30x remainder 4x - 24 - divisor * 4x0 4x - 24 remainder 0
Quotient : x2+5x+4 Remainder: 0
Trying to factor by splitting the middle term
1.5 Factoring x2+5x+4
The first term is, x2 its coefficient is 1 .
The middle term is, +5x its coefficient is 5 .
The last term, "the constant", is +4
Step-1 : Multiply the coefficient of the first term by the constant 1 • 4 = 4
Step-2 : Find two factors of 4 whose sum equals the coefficient of the middle term, which is 5 .
-4 + -1 = -5 -2 + -2 = -4 -1 + -4 = -5 1 + 4 = 5 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, 1 and 4
x2 + 1x + 4x + 4
Step-4 : Add up the first 2 terms, pulling out like factors :
x • (x+1)
Add up the last 2 terms, pulling out common factors :
4 • (x+1)
Step-5 : Add up the four terms of step 4 :
(x+4) • (x+1)
Which is the desired factorization
Equation at the end of step1: (x + 4) • (x + 1) • (x - 6) = 0