Question

X³+13x²+32x+20factorise​

Answer 1

насколько мне позволяет знание этого языка, вам надо разложить на множители данный многочлен.

x³+13x²+32x+20

Проверкой легко убедиться, что корнями являются числа -1;-2;-10, т.к. корни ищем путем перебора делителей свободного члена, т.е. среди чисел ±1;±2;±4;±5;±10;±20.

Действительно, (-1)³+13*(-1)²+32*(-1)+20=0

(-2)³+13*(-2)²+32*(-2)+20=0

(-10)³+13*(-10)²+32*(-10)+20=0

Значит, x³+13x²+32x+20=(х+1)(х+2)(х+10)

Answer 2

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Answer 3

To factorize the expression x³ + 13x² + 32x + 20, we can follow these steps:

Step 1: Check for any common factors among the coefficients. In this case, all the coefficients are relatively prime, so there are no common factors to factor out.

Step 2: Look for any rational roots using the rational root theorem. The rational root theorem states that if a polynomial equation has a rational root p/q (where p is a factor of the constant term and q is a factor of the leading coefficient), then p must divide the constant term (20 in this case) and q must divide the leading coefficient (1 in this case).

The possible rational roots are: ±1, ±2, ±4, ±5, ±10, ±20

Step 3: Test the possible rational roots using synthetic division or polynomial long division to find any roots. Synthetic division is a quicker method, so let's use that.

Testing -1 as a potential root:

diffCopy code
-1 | 1   13   32   20
    |    -1   -12   -20
    +________________
       1   12   20    0

Since the remainder is 0, -1 is a root.

Step 4: Now, we have a root -1, so we can rewrite the polynomial as a factorized form using the factor theorem. The factor theorem states that if (x - r) is a factor of a polynomial, where r is a root of the polynomial, then the polynomial can be written as (x - r) times another polynomial.

The factorized form so far is: (x + 1)(x² + 12x + 20)

Step 5: Factorize the quadratic polynomial (x² + 12x + 20). This can be factored further by finding two numbers whose sum is 12 and whose product is 20. The numbers are 2 and 10.

The factorized form of (x² + 12x + 20) is: (x + 1)(x + 2)(x + 10)

Putting it all together, the final factorized form of the expression x³ + 13x² + 32x + 20 is: (x + 1)(x + 2)(x + 10)